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% The Project Gutenberg EBook of Some Famous Problems of the Theory of    %
% Numbers and in particular Waring's Problem, by G. H. (Godfrey Harold) Hardy
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
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% Title: Some Famous Problems of the Theory of Numbers and in particular Waring's Problem
%        An Inaugural Lecture delivered before the University of Oxford   %
%                                                                         %
% Author: G. H. (Godfrey Harold) Hardy                                    %
%                                                                         %
% Release Date: August 10, 2011 [EBook #37030]                            %
%                                                                         %
% Language: English                                                       %
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% Character set encoding: ISO-8859-1                                      %
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% *** START OF THIS PROJECT GUTENBERG EBOOK FAMOUS PROBLEMS OF THEORY OF NUMBERS ***
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The Project Gutenberg EBook of Some Famous Problems of the Theory of
Numbers and in particular Waring's Problem, by G. H. (Godfrey Harold) Hardy

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net


Title: Some Famous Problems of the Theory of Numbers and in particular Waring's Problem
       An Inaugural Lecture delivered before the University of Oxford

Author: G. H. (Godfrey Harold) Hardy

Release Date: August 10, 2011 [EBook #37030]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK FAMOUS PROBLEMS OF THEORY OF NUMBERS ***
\end{PGtext}
\clearpage
\begin{PGtext}
Produced by Anna Hall, Brenda Lewis and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/American Libraries.)
\end{PGtext}
\vfill

\BookMark{0}{Transcriber's Note}
\subsection*{\centering\normalfont\scshape
\normalsize\MakeLowercase{\TransNote}}

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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\FrontMatter
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\begin{center}

\textbf{\LARGE SOME FAMOUS PROBLEMS} \\
\vfill

of the \\[12pt]
\vfill

\textbf{\LARGE THEORY OF NUMBERS} \\
\vfill

and in particular \\[12pt]
\vfill

{\LARGE Waring's Problem} \\
\vfill
\vfill

\textit{An Inaugural Lecture delivered before the \\
University of Oxford}
\vfill

{\scriptsize BY} \\
\vfill

\textbf{G.~H. HARDY, M.A., F.R.S\@.} \\
\medskip
{\itshape\scriptsize Fellow of New College \\
Savilian Professor of Geometry in the University of Oxford \\
and late Fellow of Trinity College, Cambridge\par}
\vfill
\vfill

OXFORD \\
AT THE CLARENDON PRESS \\
1920
\end{center}
%% -----File: 002.png---
%[Blank Page]
%% -----File: 003.png---
\iffalse %%duplicate of previous page
SOME  FAMOUS  PROBLEMS
of the
THEORY OF NUMBERS
and in particular
Waring's  Problem

\textit{An Inaugural Lecture delivered before the
University of Oxford}

BY

G.~H. HARDY, M.A., F.R.S\@.

\textit{Fellow of New College
Savilian Professor of Geometry in the University of Oxford
and late Fellow of Trinity College, Cambridge}

OXFORD

AT THE  CLARENDON  PRESS

1920
\fi
%% -----File: 004.png---
\newpage
\begin{center}
\null\vfill

OXFORD UNIVERSITY PRESS \\
{\scriptsize LONDON EDINBURGH GLASGOW NEW YORK \\
TORONTO MELBOURNE CAPE TOWN BOMBAY\par}
\smallskip
HUMPHREY MILFORD \\
{\scriptsize PUBLISHER TO THE UNIVERSITY}
\vfill

\end{center}
%% -----File: 005.png---
\MainMatter
\Title{SOME FAMOUS PROBLEMS OF THE
THEORY OF NUMBERS.}

\First{It} is expected that a professor who delivers an inaugural
lecture should choose a subject of wider interest than those
which he expounds to his ordinary classes. This custom is
entirely reasonable; but it leaves a pure mathematician
faced by a very awkward dilemma. There are subjects in
which only what is trivial is easily and generally comprehensible.
Pure mathematics, I am afraid, is one of them;
indeed it is more: it is perhaps the one subject in the
world of which it is true, not only that it is genuinely
difficult to understand, not only that no one is ashamed of
inability to understand it, but even that most men are
more ready to exaggerate than to dissemble their lack of
understanding.

There is one method of meeting such a situation which
is sometimes adopted with considerable success. The
lecturer may set out to justify his existence by enlarging
upon the overwhelming importance, both to his University
and to the community in general, of the particular studies on
which he is engaged. He may point out how ridiculously
inadequate is the recognition at present afforded to them;
how urgent it is in the national interest that they should be
largely and immediately re-endowed; and how immensely
all of us would benefit were we to entrust him and his
colleagues with a predominant voice in all questions of
educational administration. I have observed friends of my
own, promoted to chairs of various subjects in various
%% -----File: 006.png---
Universities, addressing themselves to this task with an
eloquence and courage which it would be impertinent in me
to praise. For my own part, I trust that I am not lacking
in respect either for my subject or myself. But, if I am
asked to explain how, and why, the solution of the problems
which occupy the best energies of my life is of
importance in the general life of the community, I must
decline the unequal contest: I have not the effrontery to
develop a thesis so palpably untrue. I must leave it to the
engineers and the chemists to expound, with justly prophetic
fervour, the benefits conferred on civilization by
gas-engines, oil, and explosives. If I could attain every
scientific ambition of my life, the frontiers of the Empire
would not be advanced, not even a black man would be
blown to pieces, no one's fortune would be made, and least
of all my own. A pure mathematician must leave to
happier colleagues the great task of alleviating the sufferings
of humanity.

I suppose that every mathematician is sometimes depressed,
as certainly I often am myself, by this feeling of
helplessness and futility. I do not profess to have any
very satisfactory consolation to offer. It is possible that
the life of a mathematician is one which no perfectly
reasonable man would elect to live. There are, however,
one or two reflections from which I have sometimes found
it possible to extract a certain amount of comfort. In the
first place, the study of mathematics is, if an unprofitable,
a perfectly harmless and innocent occupation, and we have
learnt that it is something to be able to say that at any
rate we do no harm. Secondly, the scale of the universe
is large, and, if we are wasting our time, the waste of the
lives of a few university dons is no such overwhelming
catastrophe. Thirdly, what we do may be small, but it
has a certain character of permanence; and to have produced
%% -----File: 007.png---
anything of the slightest permanent interest, whether
it be a copy of verses or a geometrical theorem, is to have
done something utterly beyond the powers of the vast
majority of men. And, finally, the history of our subject
does seem to show conclusively that it is no such mean
study after all. The mathematicians of the past have not
been neglected or despised; they have been rewarded in a
manner, undiscriminating perhaps, but certainly not ungenerous.
At all events we can claim that, if we are
foolish in the object of our devotion, we are only in our
small way aping the folly of a long line of famous men,
and that, in these days of conflict between ancient and
modern studies, there must surely be something to be said
for a study which did not begin with Pythagoras, and will
not end with Einstein, but is the oldest and the youngest
of all.

It seemed to me for a moment, when I was considering
what subject I should choose, that there was perhaps one
which might, in a philosophic University like this, be of
wider interest than ordinary technical mathematics. If
modern pure mathematics has any important applications,
they are the applications to philosophy made by the mathematical
logicians of the last thirty years. In the sphere of
philosophy we mathematicians put forward a strictly
limited but absolutely definite claim. We do not claim
that we hold in our hands the key to all the riddles of
existence, or that our mathematics gives us a vision of
reality to which the less fortunate philosopher cannot
attain; but we do claim that there are a number of
puzzles, of an abstract and elusive kind, with which the
philosophers of the past have struggled ineffectually, and
of which we now can give a quite definite and explicit
solution. There is a certain region of philosophical territory
which it is our intention to annex. It is a strictly
%% -----File: 008.png---
demarcated region, but it has suffered under the misrule of
philosophers for generations, and it is ours by right; we
propose to accept the mandate for it, and to offer it the
opportunity of self-determination under the mathematical
flag. Such at any rate is the thesis which I hope it may
before long be my privilege to defend.

It seemed to me, however, when I considered the matter
further, that there are two fatal objections to mathematical
philosophy as a subject for an inaugural address. In the
first place the subject is one which requires a certain
amount of application and preliminary study. It is not
that it is a subject, now that the foundations have been
laid, of any extraordinary difficulty or obscurity; nor that
it demands any wide knowledge of ordinary mathematics.
But there are certain things that it does demand; a little
thought and patience, a little respect for mathematics, and
a little of the mathematical habit of mind which comes
fully only after long years spent in the company of mathematical
ideas. Something, in short, may be learnt in a
term, but hardly in a casual hour.

In the second place, I think that a professor should
choose, for his inaugural lecture, a subject, if such a subject
exists, to which he has made himself some contribution of
substance and about which he has something new to say.
And about mathematical philosophy I have nothing new to
say; I can only repeat what has been said by the men,
Cantor and Frege in Germany, Peano in Italy, Russell and
Whitehead in England, who have originated the subject
and moulded it now into something like a definite form.
It would be an insult to my new University to offer it
a watered synopsis of some one else's work. I have therefore
finally decided, after much hesitation, to take a subject
which is quite frankly mathematical, and to give a
summary account of the results of some researches which,
%% -----File: 009.png---
whether or no they contain anything of any interest or
importance, have at any rate the merit that they represent
the best that I can do.

My own favourite subject has certain redeeming advantages.
It is a subject, in the first place, in which a large
proportion of the most remarkable results are by no means
beyond popular comprehension. There is nothing in the
least popular about its \emph{methods}; as to its votaries it is the
most beautiful, so by common consent it is the most difficult
of all branches of a difficult science; but many of the
actual results are such as can be stated in a simple and
striking form. The subject has also a considerable historical
connexion with this particular chair. I do not wish to
exaggerate this connexion. It must be admitted that the
contributions of English mathematicians to the Theory of
Numbers have been, in the aggregate, comparatively slight.
Fermat was not an Englishman, nor Euler, nor Gauss, nor
Dirichlet, nor Riemann; and it is not Oxford or Cambridge,
but Göttingen, that is the centre of arithmetical research
to-day. Still, there has been an English connexion, and it
has been for the most part a connexion with Oxford and
with the Savilian chair.

The connexion of Oxford with the theory of numbers is
in the main a nineteenth-century connexion, and centres
naturally in the names of Sylvester and Henry Smith.
There is a more ancient, if indirect, connexion which I
ought not altogether to forget. The theory of numbers,
more than any other branch of pure mathematics, has
begun by being an empirical science. Its most famous
theorems have all been conjectured, sometimes a hundred
years or more before they have been proved; and they
have been suggested by the evidence of a mass of computation.
Even now there is a considerable part to be
played by the computer; and a man who has to undertake
%% -----File: 010.png---
laborious arithmetical computations is hardly likely to
forget what he owes to Briggs. However, this is ancient
history, and it is with Sylvester and Smith that I am
concerned to-day, and more particularly with Smith.

Henry Smith was very many things, but above all things
a most brilliant arithmetician. Three-quarters of the first
volume of his memoirs is occupied with the theory of
numbers, and Dr.~Glaisher, his mathematical biographer,
has observed very justly that, even when he is primarily
concerned with other matters, the most striking feature of
his work is the strongly arithmetical spirit which pervades
the whole. His most remarkable contributions to the
theory are contained in his memoirs on the arithmetical
theory of forms, and in particular in the famous memoir on
the representation of numbers by sums of five squares,
crowned by the Paris Academy and published only after
his death. This memoir is peculiarly interesting to me, for
the problem is precisely one of those of which I propose to
speak to-day; and I may perhaps add one comment on the
surprising history set out in Dr.~Glaisher's introduction.
The name of Minkowski is familiar to-day to many, even
in Oxford, who have certainly never read a line of Smith.
It is curious to contemplate at a distance the storm of
indignation which convulsed the mathematical circles of
England when Smith, bracketed after his death with the
then unknown German mathematician, received a greater
honour than any that had been paid to him in life.

The particular problems with which I am concerned
belong to what is called the `additive' side of higher
arithmetic. The general problem may be stated as follows.

Suppose that $n$~is any positive integer, and
\[
\alpha_{1},\thickspace \alpha_{2},\thickspace \alpha_{3},\thickspace \dots
\]
positive integers of some special kind, squares, for example,
%% -----File: 011.png---
or cubes, or perfect $k$th~powers, or primes. We consider all
possible expressions of~$n$ in the form
\[
n = \alpha_{1} + \alpha_{2} + \dots + \alpha_{s},
\]
where $s$~may be fixed or unrestricted, the $\alpha$'s may or may
not be necessarily distinct, and order may or may not be
relevant, according to the particular problem on which we
are engaged. We denote by
\[
r(n)
\]
the number of representations which satisfy the conditions
of the problem. Then \emph{what can we say about~$r(n)$}? Can
we find an exact formula for~$r(n)$, or an approximate formula
valid for large values of~$n$? In particular, is $r(n)$ \emph{always
positive}? Is it always possible, that is to say, to find at
least \emph{one} representation of~$n$ of the type required? Or, if
this is not so, is it at any rate always possible when $n$~is
sufficiently large?

I can illustrate the nature of the general problem most
simply by considering for a moment an entirely trivial
case. Let us suppose that there are three different~$\alpha$'s only,
viz.\ the numbers $1$,~$2$,~$3$; that repetitions of the same~$\alpha$ are
permissible; that the order of the~$\alpha$'s is irrelevant; and
that~$s$, the number of the~$\alpha$'s, is unrestricted. Then it is
easy to see that $r(n)$, the number of representations, is the
number of solutions of the equation
\[
n = x + 2y + 3z
\]
in positive integers, including zero.

There are various ways of solving this extremely simple
problem. The most interesting for our present purpose is
that which rests on an analytical foundation, and uses the
idea of the \emph{generating function}
\[
f(x) = 1 + \sum_{1}^{\infty} r(n)x^{n},
\]
%% -----File: 012.png---
in which the coefficients are the values of the arithmetical
function~$r(n)$. It follows immediately from the definition
of~$r(n)$ that
\begin{multline*}
f(x) = (1 + x + x^{2} + \dots)(1 + x^{2} + x^{4} + \dots)(1 + x^{3} + x^{6} + \dots)\\
     = \frac{1}{(1 - x)(1 - x^{2})(1 - x^{3})};
\end{multline*}
and, in order to determine the coefficients in the expansion,
nothing more than a little elementary algebra is required.
We find, by the ordinary theory of partial fractions, that
\begin{multline*}
f(x) = \frac{1}{6(1 - x)^{3}} + \frac{1}{4(1 - x)^{2}}
     + \frac{17}{72(1 - x)} + \frac{1}{8(1 + x)}\\
     + \frac{1}{9(1 - \omega x)} + \frac{1}{9(1 - \omega^{2} x)},
\end{multline*}
where $\omega$~and~$\omega^{2}$ denote as usual the two complex cube roots
of unity. Expanding the fractions, and picking out the
coefficient of~$x^{n}$, we obtain
\[
r(n) = \frac{(n + 3)^{2}}{12} - \frac{7}{72} + \frac{(-1)^{n}}{8} + \frac{2}{9} \cos \frac{2n\pi}{3}.
\]
It is easily verified that the sum of the last three terms
can never be as great as~$\frac{1}{2}$, so that $r(n)$~is the integer
nearest to
\[
\frac{(n + 3)^{2}}{12}.
\]

The problem is, as I said, quite trivial, but it is interesting
none the less. A great deal of work has been done
on problems similar in kind, though naturally far more
complex and difficult in detail, by Cayley and Sylvester,
for example, in the last century, and by Glaisher, and
above all by \DPtypo{Macmahon}{MacMahon}, in this. And even this problem,
%% -----File: 013.png---
simple as it is, has sufficient content to bring out clearly
certain principles of cardinal importance.

In particular, the solution of the problem shows quite
clearly that, if we are to attack these `additive' problems
by analytic methods, it is in the theory of integral power
series
\[
  \sum a_{n} x^{n}
\]
that the necessary machinery must be found. It is this
characteristic which distinguishes this theory sharply from
the other great side of the analytic theory of numbers, the
`multiplicative' theory, in which the fundamental idea is
that of the resolution of a number into primes. In the
latter theory the right weapon is generally not a power
series, but what is called a Dirichlet's series, a series of the
type
\[
  \sum a_{n} n^{-s}.
\]
It is easy to see this by considering a simple example.
One of the most interesting functions of the multiplicative
theory is~$d(n)$, the number of divisors of~$n$. The associated
power series
\[
 \sum d(n) x^{n}
\]
is easily transformed into the series
\[
 \sum \frac{x^{n}}{1 - x^{n}},
\]
called Lambert's series. The function is an interesting
one, but somewhat unmanageable, and certainly not one of
the fundamental functions of analysis. The corresponding
Dirichlet's series is far more fundamental; it is in fact
\[
  \sum \frac{d(n)}{n^{s}} = \left(\sum \frac{1}{n^{s}}\right)^{2}
    = \Big(\zeta (s)\Big)^{2},
\]
the square of the famous Zeta function of Riemann.
%% -----File: 014.png---

The underlying reason for this distinction is fairly
obvious. It is natural to \emph{multiply} primes and unnatural
to \emph{add} them. Now
\[
  m^{-s} × n^{-s} = (mn)^{-s},
\]
so that, in the theory of Dirichlet's series, the terms
combine naturally with one another in a `multiplicative'
manner. But
\[
  x^{m} × x^{n} = x^{m + n},
\]
so that the multiplication of two terms of a power series
involves an additive operation on their ranks. It is
thus that the Dirichlet's series rather than the power
series proves to be the proper weapon in the theory of
primes.

It would be difficult for anybody to be more profoundly
interested in anything than I am in the theory of primes;
but it is not of this theory that I propose to speak to-day,
and we must return to our general additive problem. As
soon as we try to specialize the problem in some more
interesting manner, two problems stand out as calling for
research. Each of them, naturally, is only the representative
of a class.

The first of these problems is the \emph{problem of partitions}.
Let us suppose now that the $\alpha$'s are \emph{any} positive integers,
and that (as in the trivial problem) repetitions are allowed,
order is irrelevant, and $s$~is unrestricted. The problem is
then that of expressing~$n$ in any manner as a sum of
integral parts, or of solving the equation
\[
  n = x + 2y + 3z + 4u + 5v + \dots,
\]
and $r(n)$ or, as it is now more naturally written, $p(n)$, is
the number of \emph{unrestricted partitions} of~$n$. Thus
\begin{multline*}
  5 = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 2\\
    = 1 + 2 + 2 = 1 + 1 + 3 = 2 + 3 = 1 + 4 = 5,
\end{multline*}
%% -----File: 015.png---
so that $p(5) = 7$. The generating function in this case
was found by Euler, and is
\[
  f(x) = 1 + \sum_{1}^{\infty} p(n) x^{n}
    = \frac{1}{(1 - x) (1 - x^{2}) (1 - x^{3}) \dots}.
\]

I do not wish to discuss this problem in any detail
now, but the form of the generating function calls for
one or two general remarks. In the trivial problem the
generating function was \emph{rational}, with a finite number of
poles all situated upon the unit circle. Here also we are
led to a power series, or infinite product, convergent inside
the unit circle; but there the resemblance ends. This
function will be recognized by any one familiar with the
theory of elliptic functions; it is an elliptic modular
function; and, like all such functions, it has the unit circle
as a continuous line of singularities and does not exist at
all outside. It is easy to imagine the immensely increased
difficulties of any analytic solution of the problem.

It was conjectured by a very brilliant Hungarian
mathematician, Mr.~G. Pólya, five or six years ago, that
\emph{any} function represented by a power series whose
coefficients are \emph{integers}, and which is convergent inside
the unit circle, must behave, in this respect, like one or
other of the two generating functions which we have
considered. Either such a function is a rational function,
that is to say, completely elementary; or else the unit
circle is a line of essential singularities. I believe that a
proof of this theorem has now been found by Mr.~F. Carlson
of Upsala, and is to be published shortly in the \textit{Mathematische
Zeitschrift}. It is difficult for me to give reasoned
praise to a memoir which I have not seen, but I can
recommend the theorem to your attention with confidence
as one of the most beautiful of recent years.

The problem of partitions is one to which, in collaboration
with the Indian mathematician, Mr.~S. Ramanujan, I have
%% -----File: 016.png---
myself devoted a great deal of work. The principal result
of our work has been the discovery of an approximate
formula for~$p(n)$ in which the leading term is
\[
\frac{1}{2\pi\sqrt{2}}
\frac{d}{dn}
\frac{e^{\frac{2\pi}{\sqrt{6}}\sqrt{n - \smash[b]{\frac{1}{24}}}}}
     {\sqrt{n - \smash[b]{\frac{1}{24}}}},
\]
and which enables us to approximate to~$p(n)$ with an
accuracy which is almost uncanny. We are able, for
example, by using $8$~terms of our formula, to calculate
$p(200)$, a number of $13$~figures, with an error of~$.004$.
I have set out the details of the calculation in \TableRef{Table~I}{I.}\@.
\begin{table}[hbt!]
\Table{I.}
\[
\begin{array}{r}
\ColHead{p(200)}\\
3,972,998,993,185.896\\
36,282.978\\
\Tally[-]{87.555}\\
\Tally{5.147}\\
\Tally{1.424}\\
\Tally{0.071}\\
\Tally{0.000}\\
\Tally{0.043}\\
\hline
3,972,999,029,388.004\\
\hline
\end{array}
\]
\end{table}
The value of~$p(200)$ was subsequently verified by Major
MacMahon, by a direct computation which occupied over
a month.

The formulae connected with this problem are very
elaborate, and except on the purely numerical side, where
the results of the theory are compared with those of computation,
it is not very well suited for a hasty exposition;
and I therefore pass on at once to the principal
object of my lecture, the very famous problem known,
after a Cambridge professor of the eighteenth century, as
\emph{Waring's Problem}.
%% -----File: 017.png---

We suppose now that every~$\alpha$ is a perfect $k$-th~power
$m^{k}$, $k$~being fixed in each case of the problem which we
consider; $m$~may be of either sign if $k$~is even, but must
be positive if $k$~is odd. In either case we allow $m$ to be
zero. Repetitions are permitted, as in our previous problems;
but it is more convenient now to take account of the order
of the~$\alpha$'s; and $s$, which was formerly unrestricted, is now
fixed in each case of the problem, like~$k$. The problem is
therefore that of determining the number of representations
of a number~$n$ as the sum of $s$~positive $k$-th~powers. Thus
Henry Smith's problem, the problem of five squares, is the
particular case of Waring's problem in which $k$~is~$2$ and
$s$~is~$5$. The problem has a long history, which centres
round this simplest case of squares; a history which began,
I suppose, with the right-angled triangles of Pythagoras,
and has been continued by a long succession of mathematicians,
including Fermat, Euler, Lagrange, and Jacobi, down
to the present day. I will begin by a summary of what
is known in the simplest case, where the solution is
practically complete.

A number~$n$ is the sum of \textbf{two} squares if and only if
it is of the form
\[
n = M^{2}P,
\]
where $P$~is a product of primes, all different and all of the
form~$4k + 1$. In particular, a prime number of the form
$4k + 1$ can be expressed as the sum of two squares, and
substantially in only one way. Thus $5 = 1^{2} + 2^{2}$, and there
is no other solution except the solutions $(±1)^{2} + (±2)^{2}$,
$(±2)^{2} + (±1)^{2}$, which are not essentially different, although
it is convenient to count them as distinct. The number of
numbers less than~$x$, and expressible as the sum of two
squares, is approximately
\[
  \frac{Cx}{\sqrt{\log x}},
\]
%% -----File: 018.png---
where $C$~is a certain constant. The last result was proved
by Landau in~1908; all the rest belong to the classical
theory.

A number is the sum of \textbf{three} squares unless it is of
the form
\[
4^{\alpha}(8k + 7),
\]
when it is not so expressible. \emph{Every} number may be
expressed by \textbf{four} squares, and \textit{a~fortiori} by five or more.
It is this last theorem of Lagrange that I would ask you
particularly to bear in mind.

If $s$, the number of squares, is even and less than~$10$,
the number of representations may be expressed in a very
simple form by means of the divisors of~$n$. Thus the
number of representations by $4$~squares, when $n$~is odd, is
$8$~times the sum of the divisors of~$n$; when $n$~is even, it
is $24$~times the sum of the odd divisors; and there are
similar results for $2$~squares, or~$6$, or~$8$.

When $s$~is $3$,~$5$, or~$7$, the number of representations can
also be found in a simple form, though one of a very
different character. Suppose, for example, that $s$~is~$3$. The
problem is in this case essentially the same as that of
determining the number of classes of binary arithmetical
forms of determinant~$-n$; and the solution depends on
certain finite sums of the form
\[
\sum \beta,\quad \sum \gamma,
\]
extended over quadratic residues~$\beta$ or non-residues~$\gamma$ of~$n$.

When $s$, whether even or odd, is greater than~$8$, the
solution is decidedly more difficult, and it is only very
recently that a uniform method of solution, for which I
must refer you to some recent memoirs of Mr.~L.~J. Mordell
and myself, has been discovered. For the moment I wish
to concentrate your attention on two points: the first, that
%% -----File: 019.png--- %%moved up one line to avoid breaking emph.
\emph{an expression by $4$~squares is always possible, while one
by~$3$ is not}; and the second, that the existence of numbers
not expressible by $3$~squares is revealed at once by the quite
trivial observation that no number so expressible can be
congruent to~$7$ to modulus~$8$.

It is plain, when we proceed to the general case, that
any number~$n$ can be expressed as a sum of $k$-th~powers;
we have only to take, for example, the sum of $n$~ones. And,
when $n$~is given, there is a \emph{minimum} number of $k$-th~powers
in terms of which $n$~can be expressed; thus
\[
23 = 2·2^{3} + 7·1^{3}
\]
is the sum of $9$~cubes and of no smaller number. But it is
not at all plain (and this is the point) that this minimum
number cannot tend to infinity with~$n$. It does not when
$k = 2$; for then it cannot exceed~$4$. And Waring's Problem
(in the restricted sense in which the name has commonly
been used) is the problem of proving that the minimum
number is similarly bounded in the general case. It is not
an easy problem; its difficulty may be judged from the fact
that it took 127~years to solve.

We may state the problem more formally as follows.
Let $k$ be given. Then there may or may not exist a
number~$m$, the same for all values of~$n$, and such that
$n$~can always be expressed as the sum of~$m$ $k$-th~powers or
less. If any number~$m$ possesses this property, all larger
numbers plainly possess it too; and among these numbers
we may select the \emph{least}. This least number, which will
plainly depend on~$k$, we call~$g(k)$; thus $g(k)$ is, by definition,
the least number, if such a number exists, for which
it is true that
\begin{quotation}
\textit{`every number is the sum of~$g(k)$ $k$-th~powers or less'.}
\end{quotation}
We have seen already that $g(2)$~exists and has the value~$4$.

In the third edition of his \textit{Meditationes Algebraicae},
published in Cambridge in~1782, Waring asserted that
%% -----File: 020.png---
every number is the sum of not more than $4$~squares, not
more than $9$~cubes, not more than $19$~fourth powers, \textit{et sic
deinceps}. A little more precision would perhaps have
been desirable; but it has generally been held, and I do
not question that it is true, that what Waring is asserting
is precisely the existence of~$g(k)$. He implies, moreover,
that $g(2) = 4$ and $g(3) = 9$; and both of these assertions
are correct, though in the first he had been anticipated by
Lagrange. Whether $g(4)$ is or is not equal to~$19$ is not
known to-day.

Waring advanced no argument of any kind in support
of his assertion, and it is in the highest degree unlikely
that he was in possession of any sort of proof. I have no
desire to detract from the reputation of a man who was
a very good mathematician if not a great one, and who
held a very honourable position in a University which not
even Oxford has persuaded me entirely to forget. But
there is a tendency to exaggerate the profundity implied
by the enunciation of a theorem of this particular kind.
We have seen this even in the case of Fermat, a mathematician
of a class to which Waring had not the slightest
pretensions to belong, whose notorious assertion concerning
`Mersenne's numbers' has been exploded, after the lapse of
over 250~years, by the calculations of the American computer
Mr.~Powers. No very laborious computations would be
necessary to lead Waring to a highly plausible speculation,
which is all I take his contribution to the theory to be;
and in the Theory of Numbers it is singularly easy to
speculate, though often terribly difficult to prove; and it
is only proof that counts.

The next advance towards the solution of the problem
was made by Liouville, who established the existence of~$g(4)$.
Liouville's proof, which was first published in~1859,
is quite simple and, as the simplest example of an important
%% -----File: 021.png---
type of argument, is worth reproducing here. It may be
verified immediately that
\begin{multline*}
6X^{2} = 6(x^{2} + y^{2} + z^{2} + t^{2})^{2}\\
\begin{aligned}
&= (x + y)^{4} + (x - y)^{4} + (z + t)^{4} + (z - t)^{4}\\
&+ (x + z)^{4} + (x - z)^{4} + (t + y)^{4} + (t - y)^{4}\\
&+ (x + t)^{4} + (x - t)^{4} + (y + z)^{4} + (y - z)^{4};
\end{aligned}
\end{multline*}
and since, by Lagrange's theorem, any number~$X$ is the
sum of $4$~squares, it follows that any number of the form~$6X^{2}$
is the sum of $12$~biquadrates. Hence any number of
the form $6(X^{2} + Y^{2} + Z^{2} + T^{2})$ or, what is the same thing,
any number of the form~$6m$, is the sum of $48$~biquadrates.
But \emph{any} number~$n$ is of the form~$6m + r$, where $r$~is
$0$,~$1$, $2$, $3$, $4$, or~$5$. And therefore $n$~is, at worst, the sum
of $53$~biquadrates. That is to say, $g(4)$~exists, and does
not exceed~$53$. Subsequent investigators, refining upon this
argument, have been able to reduce this number to~$37$; the
final proof that $g(4) \leq 37$, the most that is known at present,
was given by Wieferich in~1909. The number
\[
79 = 4·2^{4} + 15·1^{4}
\]
needs $19$~biquadrates, and no number is known which needs
more. There is therefore still a wide margin of uncertainty
as to the actual value of~$g(4)$.

The case of cubes is a little more difficult, and the
existence of~$g(3)$ was not established until 1895, when
Maillet proved that $g(3) \leq 17$. The proof then given by
Maillet rests upon the identity
\begin{multline*}
6x(x^{2} + y^{2} + z^{2} + t^{2})\\
= (x + y)^{3} + (x - y)^{3} + (x + z)^{3} + (x - z)^{3} + (x + t)^{3} + (x - t)^{3},
\end{multline*}
and the known results concerning the expression of a
number by $3$~squares. It has not the striking simplicity
of Liouville's proof; but it has enabled successive investigators
to reduce the number of cubes, until finally Wieferich,
%% -----File: 022.png---
in~1909, proved that $g(3) \leq 9$. As $23$~and~$239$ require
$9$~cubes, the value of~$g(3)$ is in fact exactly~$9$. It is only
for $k = 2$ and $k = 3$ that the actual value of~$g(k)$ has
been determined. But similar existence proofs were found,
and upper bounds for~$g(k)$ determined, by various writers,
in the cases $k = 5$,~$6$, $7$, $8$, and~$10$.

Before leaving the problem of the cubes I must call your
attention to another very beautiful theorem of a slightly
different character. The numbers $23$~and~$239$ require
$9$~cubes, and it appears, from the results of a survey of
all numbers up to~$40,000$, that no other number requires so
many. It is true that this has not actually been proved;
but it \emph{has} been proved (and this is of course the essential
point) that the number of numbers which require as many
cubes as~$9$ is \emph{finite}.

This singularly beautiful theorem, which is due to my
friend Professor Landau of Göttingen, and is to me as
fascinating as anything in the theory, also dates from
1909, a year which stands out for many reasons in the
history of the problem. It is of exceptional interest not
only in itself but also on account of the method by which
it was proved, which utilizes some of the deepest results in
the modern theory of the asymptotic distribution of primes,
and made it, until very recently, the only theorem of its
kind erected upon a genuinely transcendental foundation.
To me it has a personal interest also, as being the only
theorem of the kind which, up to the present, defeats
the new analytic method by which Mr.~Littlewood and
I have recently studied the problem.

Landau's theorem suggests the introduction of another
function of~$k$, which I will call~$G(k)$, of the same general
character as~$g(k)$, but I think probably more fundamental.
This number~$G(k)$ is defined as being the least number for
which it is true that
%% -----File: 023.png---
\begin{quotation}
\textit{`every \DPtypo{member}{number} \textsc{from a certain point onwards} is the
sum of~$G(k)$ $k$-th~powers or less.'}
\end{quotation}
It is obvious that the existence of~$g(k)$ involves that of~$G(k)$,
and that $G(k) \leq g(k)$. When $k = 2$, both numbers
are~$4$; but $G(3) \leq 8$, by Landau's theorem, while $g(3) = 9$;
and doubtless $G(k) < g(k)$ in general. It is important also
to observe that, conversely, the existence of~$G(k)$ involves
that of~$g(k)$. For, if $G(k)$~exists, all numbers beyond a
certain limit~$\gamma$ are sums of~$G(k)$ $k$-th~powers or less. But
all numbers less than~$\gamma$ are sums of $\gamma$~ones or less, and therefore
$g(k)$ certainly cannot exceed the greater of $G(k)$~and~$\gamma$.

I said that $G(k)$ seemed to me the more fundamental of
these numbers, and it is easy to see why. Let us assume
(as is no doubt true) that the only numbers which require
$9$~cubes for their expression are $23$~and~$239$. This is a very
curious fact which should be interesting to any genuine
arithmetician; for it ought to be true of an arithmetician
that, as has been said of Mr.~Ramanujan, and in his case
at any rate with absolute truth, that `every positive integer
is one of his personal friends'. But it would be absurd to
pretend that it is one of the profounder truths of higher
arithmetic: it is nothing more than an entertaining arithmetical
fluke. It is Landau's~$8$ and not Wieferich's~$9$ that
is the profoundly interesting number.

The real value of~$G(3)$ is still unknown. It cannot be
less than~$4$; for every number is congruent to~$0$, or~$1$, or~$-1$
to modulus~$3$, and it is an elementary deduction that
every cube is congruent to~$0$, or~$1$, or~$-1$ to modulus~$9$.
From this it follows that the sum of three cubes cannot be
of the form $9m + 4$~or $9m + 5$: for such numbers at least
$4$~cubes are necessary, so that $G(3) \geq 4$. But whether
$G(3)$~is $4$,~$5$, $6$, $7$, or~$8$ is one of the deepest mysteries
of arithmetic.

It is worth while to glance at the evidence of computation.
%% -----File: 024.png---
Dase, at the instance of Jacobi, tabulated the minimum
number of cubes for values of~$n$ from~$1$ to~$12,000$, and Daublensky
von~Sterneck has extended the table to~$40,000$.
Some of the results are shown in \TableRef{Table~II}{II.}\@.
\begin{table}[hbt!]
\Table{II.}
\[
\begin{array}{rrrrrrrrrc}
& \ColHead{1}& \ColHead{2} & \ColHead{3} & \ColHead{4} & \ColHead{5} & \ColHead{6} & \ColHead{7} & \ColHead{8} & \ColHead{9} \\
    1\EnDash\phantom{0}1000& 10& 41& 122& 242& 293& 202&      73&       15&        2\\
 1000\EnDash\phantom{0}2000&  2& 27& 113& 283& 358& 194&      23&  \EmDash&  \EmDash\\
 9000\EnDash10000&  1& 17& 121& 377& 401&  83& \EmDash&  \EmDash&  \EmDash\\
19000\EnDash20000&  1& 12& 100& 400& 426&  61& \EmDash&  \EmDash&  \EmDash\\
29000\EnDash30000&  1& 11& 105& 448& 388&  47& \EmDash&  \EmDash&  \EmDash\\
39000\EnDash40000&  1& 13& 117& 457& 384&  28& \EmDash&  \EmDash&  \EmDash
\end{array}
\]
\end{table}
In each row I have shown a typical thousand numbers,
classified according to the minimum number of cubes by
which they can be expressed. There are $15$~numbers only
for which $8$~are needed, the largest being~$454$. There are
$121$ for which $7$~are needed, the two largest being $5818$ and
$8042$; the distribution of these $121$~numbers in the first
$9$~thousands is
\[
  73,\thickspace 23,\thickspace 7,\thickspace 6,\thickspace 7,\thickspace 4,\thickspace 0,\thickspace 0,\thickspace 1.
\]
If empirical evidence means anything, it seems clear that
$G(3) \leq 6$. I am sure that Professor Townsend and Professor
Lindemann have made countless generalizations on evidence
far less substantial.

It is also clear that, throughout von~Sterneck's tables,
there is a fairly steady, though latterly very slow, decrease
in the proportion of numbers for which even $6$~cubes are
required; but that the table is not sufficiently extensive to
give any very decisive indication as to whether these numbers
disappear or not. It seemed to me this was a case in
which further evidence would be worth having. To calculate
a \emph{systematic} table on the scale required would be a work
of years. It is possible, however, to obtain some indication
%% -----File: 025.png---
of the probable truth, without any superhuman patience,
by exploring a selected stratum of much larger numbers.
Dr.~Ruckle of Göttingen recently undertook this task at
my request, and I am glad to be able to tell you his
results. He found, for the $2,000$~numbers immediately
below $1,000,000$, the following distribution.
\[
\begin{array}{lccccccc}
 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
998000\EnDash  999000 & 0 & 1 & 98 & 640 & 262 & 1 & 0\\%[**TN: there is an error in this row, as it adds up to 1002 not 1000]
999000\EnDash 1000000 & 1 & 1 & 94 & 614 & 289 & 1 & 0\\
\end{array}
\]
You will observe that the $6$-cube numbers have all but
disappeared, and that there is a quite marked turnover
from $5$ to~$4$. Conjecture in such a matter is extremely
rash, but I am on the whole disposed to predict with some
confidence that $G(3) \leq 5$. If I were asked to choose between
$5$~and~$4$, all I could say would be this. That $G(3)$ should
be~$4$ would harmonize admirably, so far as we can see
at present, with the general trend of Mr.~Littlewood's and
my researches. But it is plain that, if the $5$-cube numbers
too do ultimately disappear, it can only be among numbers
the writing of which would tax the resources of the decimal
notation; and at present we cannot \emph{prove} even that $G(3) \leq 7$,
though here success seems not impossible.

With the fourth powers or biquadrates we have been
very much more successful. I have explained that $g(4)$~lies
between $19$~and~$37$. As regards~$G(4)$, we have here no
numerical evidence on the same scale as for cubes. Any
fourth power is congruent to $0$~or~$1$ to modulus~$16$, and
from this it follows that no number congruent to~$15$ to
modulus~$16$ can be the sum of less than~$15$ fourth powers.
Thus $G(4) \geq 15$; and Kempner, by a slight elaboration of
this simple argument, has proved that $G(4) \geq 16$. No
better upper bound was known before than the~$37$ of
Wieferich, but here Mr.~Littlewood and I have been able
%% -----File: 026.png---
to make a very substantial improvement, first to~$33$ and
finally to~$21$. Thus $G(4)$~lies between $16$~and~$21$, and
the margin is comparatively small.

I turn now to the general case. In the years up to
1909, the existence proof was effected, and upper bounds
for~$g(k)$ determined, for the values of~$k$ from~$2$ to~$8$ inclusive
and for $k = 10$. These upper bounds are shown
in the first row of \TableRef{Table~III}{III.}; that for~$10$, which is not
included, is somewhere in the neighbourhood of~$140,000$.
\begin{table}[hbt!]
\Table{III.}
\[
\begin{array}{@{}l@{\ }cccrrrr}
& 2&    3&    4& \ColHead{5}& \ColHead{6}& \ColHead{7}& \ColHead{8}\\
g(k) \leq&
  4&    9&   37&   58&   478&   3806&  31353\\
\DPtypo{G(k)}{g(k)} \geq \bigl[\left(\frac{3}{2}\right)^{k}\bigr] + 2^{k} - 2 =&
  4&    9&   19&   37&    73&    143&      279\\
G(k) \leq&
   4& [8]&   37&   58&   478&   3806&  31353\\
G(k) \leq (k - 2) 2^{k-1} + 5 =&
 (5)& (9)&   \mathbf{21}&   \mathbf{53}&   \mathbf{133}&    \mathbf{325}&    \mathbf{773}\\
G(k) \geq k + 1, 4k&
   4&   4&   16&    6&     7&      8&     32
\end{array}
\]
\end{table}
In the second row I have shown the best known lower
bounds, which are given by the simple general formula
which stands to the left, in which $\left[\left(\frac{3}{2}\right)^{k}\right]$~denotes the
integral part of~$\left(\frac{3}{2} \right)^{k}$. It is easily verified, in fact, that
the number
\[
\left(\left[\left(\tfrac{3}{2}\right)^{k}\right] - 1\right) 2^{k} + 2^{k} - 1,
\]
which is less than~$3^{k}$, requires the number of $k$-th~powers
stated.\footnote
  {This observation was made by Bretschneider in~1853.}
It will be observed that the first three numbers
are those which occur in Waring's enunciation.

Waring's problem, as I have defined it---the problem,
that is to say, of finding a general existence proof
for~$g(k)$, and \textit{a~fortiori} for~$G(k)$---was ultimately solved
by Hilbert, once more in~1909. I wish that I had time
to give a proper account of his justly famous memoir,
which raised the whole discussion at once on to an
%% -----File: 027.png---
altogether higher level. As it is, I must confine myself
to one or two extremely inadequate remarks. The
proof falls into two parts. The first part is what I may
call semi-transcendental. It is not fully transcendental in
the sense in which, for example, the classical proofs in the
theory of the distribution of primes are transcendental, for
it does not make use of the machinery of the theory of
analytic functions of a complex variable; but it uses the
methods of the integral calculus, and is therefore not
fully elementary. Hilbert set out with what would appear
at first sight to be the singularly ill-adapted weapon of
a volume integral in space of $25$~dimensions, a number
which he was afterwards able to reduce to~$5$. The formula
which he ultimately used is
\begin{multline*}
(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2})^{k}\\
=  C\iiiiint (x_{1}t_{1} + x_{2}t_{2} + x_{3}t_{3} + x_{4}t_{4} +
             x_{5}t_{5})^{2k}\, dt_{1} \dots dt_{5},
\end{multline*}
where $C$~is a certain constant, viz.\
\[
\frac{(2k + 1)(2k + 3)(2k + 5)}{8\pi^{2}},
\]
and the integration is effected over the interior of the
hypersphere
\[
t_{1}^{2} + t_{2}^{2} + t_{3}^{2} + t_{4}^{2} + t_{5}^{2} = 1.
\]

Starting from this formula he was able, by an exceedingly
ingenious process based upon the definition of a
definite integral as the limit of a finite sum, to prove
the existence in the general case of algebraical identities
analogous to that used by Liouville and his followers
when $k$~is~$4$. It should be observed that Hilbert's
proof is essentially an \emph{existence proof}; his method is
not effective for the actual determination of these identities
%% -----File: 028.png---
even in the simplest cases. The identities which are known
for special values of~$k$ have been obtained by common
algebra, and are, after the first few values of~$k$, excessively
complicated. The simplest known identity for $k = 10$, for
instance, is
\begin{multline*}
22680(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})^{5}\\
\begin{aligned}
&= 9\sum^{(8)} (x_{1} ± x_{2} ± x_{3} ± x_{4})^{10}
  + \sum^{(48)} (2x_{1} ± x_{2} ± x_{3})^{10}\\
&+ 180\sum^{(12)} (x_{1} ± x_{2})^{10}
  + 9\sum^{(4)} (2x_{1})^{10},
\end{aligned}
\end{multline*}
where the figures in brackets show the number of terms
under the signs of summation. However, the identities
exist; and it should be clear to you, after our discussion of
the case $k = 4$, that they enable us at once to obtain a
proof in succession for $k = 2$,~$4$, $8$, $16$,~$\dots$ and generally
whenever $k$~is a power of~$2$. This concludes the first and
most characteristic part of Hilbert's argument. The second
part, in which the conclusion is extended to every value of~$k$,
is purely algebraical.

Hilbert's work has been reconsidered and simplified by
a number of writers, most notably by Dr.~Stridsberg of
Stockholm, and the ultimate result of their work has been
to eliminate the transcendental elements from the proof
entirely. The proof, as they have left it, is fully elementary;
it does not involve any reference to integrals, or to
any kind of limiting process, but depends simply on an
ingenious system of equations derived by the processes of
finite algebra. It remains a pure existence proof, and
throws no light on the value of~$g(k)$.

It would hardly be possible for me to exaggerate the
admiration which I feel for the solution of this historic
problem of which I have been compelled to give so bald
%% -----File: 029.png---
and summary a description. Within the limits which it
has set for itself, it is absolutely and triumphantly successful,
and it stands with the work of Hadamard and de~la
Vallée-Poussin, in the theory of primes, as one of the landmarks
in the modern history of the theory of numbers.
But there is an enormous amount which remains to be
done, and it would seem that, if we are to interpret
Waring's problem in the widest possible sense, if we are to
get into real contact with the actual values of our numbers
$g(k)$~and~$G(k)$, still more if we are to attack all the obvious
problems connected with the number of representations,
then essentially different and inherently more powerful
methods are required. There is one armoury only in which
such more powerful weapons can be found, that of the
modern theory of functions. In short we must learn how
to apply Cauchy's Theorem to the problem, and that is
what Mr.~Littlewood and I have set out to do.

The first step is fairly obvious. The formulae are
slightly simpler when $k$~is \emph{even}. The number of representations
of~$n$ as the sum of~$s$ $k$\DPtypo{}{-}th~powers, which we may
denote in general by
\[
r_{k, s}(n),
\]
is then the coefficient of~$x^{n}$ in the generating function
\[
1 + \sum_{1}^{\infty} r_{k, s}(n)x^{n} = \left(f(x)\right)^{s},
\]
where
\[
f(x) = 1 + 2x^{1^{k}} + 2x^{2^{k}} + 2x^{3^{k}} + \dots\DPtypo{}{.}
\]
This formula involves certain conventions as to the order
and sign of the numbers which occur in the representations
which are to be reckoned as distinct; but the complications
so introduced are trivial and I need not dwell on them.
The series is convergent when $|x| < 1$, and, by Cauchy's
%% -----File: 030.png---
Theorem, we have
\[
r_{k,s}(n) = \frac{1}{2\pi i} \int \frac{\left(f(x)\right)^{s}}{x^{n+1}}\, dx,
\]
the path of integration being a circle whose centre is at
the origin and whose radius is less than unity.

All this is simple enough; but the further study of the
integral is very intricate and difficult, and I cannot attempt
to do more than to give a rough idea of the obstacles that
have to be surmounted. Let us contrast the integral for a
moment with that which would stand in its place in the
`trivial' problem to which I referred early in my lecture.
There the subject of integration would be a \emph{rational} function,
with a finite number of poles all situated on the unit
circle. We could deform the contour into one which lies
wholly at a considerable distance from the origin and in
which, owing to the factor~$x^{n+1}$ in the denominator, every
element is very small when $n$~is large. We should have,
of course, to introduce corrections corresponding to the
residues at the poles; and it is just these corrections which
would give the dominant terms of an approximate formula
by means of which our coefficients could be studied. In
the present case we have no such simple recourse; for
every point of the unit circle is a singularity of an exceedingly
complicated kind, and the circle as a whole is a
barrier across which it is impossible to deform the contour.
It is of course for this reason that no successful application
of the method has been made before.

Our fundamental idea for overcoming the difficulty is
as follows. Among the continuous mass of singularities
which covers up the circle, it is possible to pick out a class
which to a certain extent dominates the rest. These special
singularities are those associated with the \emph{rational} points
of the circle, that is to say, the points
\[
x = e^{2p\pi i/q},
\]
%% -----File: 031.png---
where $p/q$~is a rational fraction in its lowest terms. This
class of points is indeed an \emph{infinite} class; but the infinity
is, in Cantor's phrase, only an \emph{enumerable} infinity; and
the points can therefore be arranged in a simply infinite
series, on the model of the series
\[
\tfrac{0}{1},\thickspace \tfrac{1}{2},\thickspace \tfrac{1}{3},\thickspace \tfrac{2}{3},\thickspace \tfrac{1}{4},\thickspace
\tfrac{3}{4},\thickspace \tfrac{1}{5},\thickspace \tfrac{2}{5},\thickspace \tfrac{3}{5},\thickspace \tfrac{4}{5},\thickspace
\tfrac{1}{6},\thickspace \tfrac{5}{6},\thickspace \tfrac{1}{7},\thickspace \dots.
\]
In the neighbourhood of these points the behaviour of the
function is\DPtypo{,}{} sufficiently complex indeed, but simpler than
elsewhere. The function has, to put the matter in a rough
and popular way, a general tendency to become large in
the neighbourhood of the unit circle, but this tendency is
most pronounced near these particular points. They are
not only the \emph{simplest} but also the \emph{heaviest} singularities;
their weight is greatest when the denominator~$q$ is smallest,
decreases as $q$~increases, and (as a physicist would say)
becomes infinitely small when $q$~is infinitely large. There
is, therefore, at any rate, the hope that we may be able to
isolate the contributions of each of these selected points,
and obtain, by adding them together, a series which may
give a genuine approximation to our coefficient.

I owe to Professor Harald Bohr of Copenhagen a
picturesque illustration which may help to elucidate the
general nature of our argument. Imagine the unit circle
as a thin circular rail, to which are attached an infinite
number of small lights of varying intensity, each illuminating
a certain angle immediately in front of it. The
brightest light is at $x = 1$, corresponding to $p = 0$, $q = 1$;
the next brightest at $x = -1$, corresponding to $p = 1$,
$q = 2$; the next at $x = e^{2\pi i/3}$ and $e^{4\pi i/3}$, and so on. We
have to arrange the inner circle, the circle of integration,
in the position of maximum illumination. If it is too far
away the light will not reach it; if too near, the arcs
which fall within the angles of illumination will be too
%% -----File: 032.png---
narrow, and the light will not cover it completely. Is it
possible to place it where it will receive a satisfactorily
uniform illumination?

The answer is that this is \emph{only} possible when $k$~is~$2$.
Our functions are then elliptic functions; the lights are the
formulae of the theory of linear transformation; and we
can find a position of the inner circle in which it falls
entirely under their rays. We are thus led to a solution of
the problem of the squares which is in all essential respects
complete. But when $k$~exceeds~$2$ the result is less satisfactory.
The angle of the lights is then too narrow; the
beams which they emit, instead of spreading out with
reasonable regularity, are shaped like torpedoes or cigars;
however we move our circle a part remains in darkness.
It would seem that this difficulty, which held up our
researches for something like two years, is the really
characteristic difficulty of the general problem. It cannot
be solved until we have found some other source of light.

It was only after the most prolonged and painful efforts
that we were able to discover such another source. It is
possible not only to hang lights upon the rail, but also, to
a certain extent, to cause the rail itself to glow. The
illumination which can be induced in this manner is irritatingly
faint, and it is for this reason that our results are
not yet all that we desire; but it is enough to make the
dark places dimly visible and to enable us to prove a great
deal more than has been proved before.

The actual results which we obtain are these. We find
that there is a certain series, which we call the \emph{singular
series}, which is plainly the key to the solution. This
series is
\[
\S = \sum \left(\frac{S_{p,q}}{q}\right)^{s} e^{2np\pi i/q},
\]
%% -----File: 033.png---
where
\[
S_{p,q} = \sum_{h=0}^{q-1} e^{2h^{k}p\pi i/q}
\]
---a sum which reduces, when $k = 2$, to one of what are
known as `Gauss's sums'---and the summation extends,
first to all values of $p$ less than and prime to~$q$, and
secondly to all positive integral values of~$q$. The genesis
of the series is this. We associate with the rational point
$x = e^{2p\pi i/q}$ an auxiliary power series
\[
f_{p,q}(x)= \sum_{n} c_{p,q,n}x^{n},
\]
which (\textit{a})~is as simple and natural as we can make it, and
(\textit{b})~behaves perfectly regularly at all points of the unit circle
except at the one point with which we are particularly concerned.
We then add together all these auxiliary functions,
and endeavour to approximate to the coefficient of our original
series by summing the auxiliary coefficients over all values
of $p$~and~$q$. The process is, at bottom, one of `decomposition
into simple elements', applied in an unusual way.

Our final formula for the number of representations is
\[
r_{k,s}(n) =
  \frac{\left\{2\Gamma\Bigl(1 + \dfrac{1}{k}\Bigr)\right\}^{s}}
       {\Gamma\Bigl(1 + \dfrac{s}{k}\Bigr)}
        n^{\frac{s}{k}-1} \S + O(n^{\sigma}),
\]
the second term denoting an error less than a constant
multiple of~$n^{\sigma}$, and $\sigma$~being a number which is less than
$\dfrac{s}{k} - 1$ at any rate for sufficiently large values of~$s$. The
second term is then of lower order than the first. Further,
the first term is real, and it may be shown, if $s$~surpasses
a certain limit, to be \emph{positive}. If both these conditions
are satisfied, and $n$~is sufficiently large, then $r_{k,s}(n)$
%% -----File: 034.png---
cannot be zero, and representations of~$n$ by $s$ $k$\DPtypo{}{-}th~powers
certainly exist. The way is thus open to a proof of the
existence of~$G(k)$; if $G(k)$~exists, so also does~$g(k)$, and
Waring's problem is solved.

The structure of the dominant term in our general
formula is best realized by considering some special cases.
In \TableRef{Table~IV}{IV.} I have written out the leading terms of~$\S$,
\begin{table}[hbt!]
\Table{IV.}
\[
\begin{array}{@{}l}
\ColHead{k = 2.}\\
\S = 1 + 0 +
  \dfrac{2}{3^{\frac{1}{2}s}}
    \cos\left(\dfrac{2}{3} n\pi - \dfrac{1}{2} s\pi\right) +
  \dfrac{2^{\frac{1}{2}s+1}}{4^{\frac{1}{2}s}}
    \cos\left(\dfrac{1}{2}n\pi - \dfrac{1}{4} s\pi\right)\\[3ex]
  \hspace*{3cm}+ \dfrac{2}{5^{\frac{1}{2} s}}
        \left\{\cos\dfrac{2}{5} n\pi +
                 \cos\left(\dfrac{4}{5} n\pi - s\pi\right)\right\} +
                   0 + \dots.\\[4ex]
\ColHead{k = 3, s = 7.}\\[1ex]
\S = 1 + 0.610 \cos \frac{2}{9} n\pi + 0.130 \cos \frac{2}{7} n\pi +
         0.078 \cos \frac{6}{7} n\pi + \dots.\\[3ex]
\ColHead{k = 4, s = 33.}\\[1ex]
\S = 1 + 1.054 \cos (\frac{1}{8} n\pi - \frac{1}{16} \pi) +
       0.147 \cos (\frac{1}{4} n\pi - \frac{1}{8} \pi) + \dots.\\[3ex]
\ColHead{k = 4, s = 21.}\\[1ex]
\S = 1 + 1.331 \cos (\frac{1}{8} n\pi + \frac{11}{16} \pi) +
           0.379 \cos (\frac{1}{4} n\pi - \frac{5}{8} \pi) + \dots.
\end{array}
\]
\end{table}
first when $k = 2$ and $s$~is arbitrary, and then for $7$~cubes
and for $33$~and $21$~biquadrates. There are certain
characteristics common to all these series. The terms
diminish rapidly; in each case only a very few are of real
importance: and they are oscillatory, with a period which
increases as the amplitude of the oscillations decreases.
The series for the cubes is easily shown to be positive;
but we cannot deduce that $r_{3,7}(n)$ is positive, and draw
consequences as to the representation of numbers by $7$~cubes,
%% -----File: 035.png---
because in this case we cannot dispose satisfactorily of the
error term~$O(n^{\sigma})$ in the general formula. In the two
cases relating to fourth powers which I have chosen, the
discussion of the series itself is rather more delicate, for
there is in each of them one term which can be negative
and greater than~$1$. But the discussion can be brought to
a satisfactory conclusion, and, as in this case we are able
to prove that the error term is really of lower order, we
obtain what we desire. \emph{Every large number is the sum of
$21$~fourth powers or less}; $G(4) \leq 21$. Further, we have
obtained a genuine asymptotic formula for the number
of representations, which can be used for the study of the
representations of numbers of particular forms. We can
show, for example, that a large number of the form
$16n + 10$ can be expressed by $21$~biquadrates in about $200$~times
more ways than one of the form~$16n + 2$.

If the method of which I have tried to give some general
idea is compared with those which have previously been
applied to the problem, it will be found that it has three
very great advantages. In the first place it is inherently
very much more powerful. It brings us for the first time
into relation with the series on which the solution in the
last resort depends, and tells us, approximately but truly,
what the number of representations really is. Secondly, it
gives us numerical results which, as soon as $k$~exceeds~$3$, are
far in advance of any known before. These numbers
are those in the fourth row of \TableRef{Table~III}{III.}\@.\footnote
  {The thick type indicates a new result. The (5)~and~(9) in round
  brackets are inferior to results already known. Our method is easily
  adapted to deal with the case $k = 2$ completely; but it will not at
  present yield Landau's~$8$, which is therefore enclosed in square
  brackets.}
It will be seen
that these numbers conform to a simple law, and that is
the third advantage of the method, that it is not a mere
%% -----File: 036.png---
existence proof, but gives us a definite upper bound for~$G(k)$
for all values of~$k$, viz.\
\[
G(k) \leq (k - 2) 2^{k-1} + 5.
\]

In the last row of the table~I have shown all that is known
about~$G(k)$ on the other side. In all cases $G(k) \geq k + 1$,
while if $k$~is a power of~$2$ we can say more, namely that
$G(k) \geq 4k$. A comparison between this row of figures and
that above it is enough to show the room which remains
for further research. It is beyond question that our
numbers are still very much too large; and there is no
sort of finality about our researches, for which the best
that we claim is that they embody a method which opens
the door for more.

I will conclude by one word as to the application of
our method to another and a still more difficult problem.
It was asserted by Goldbach in 1742 that \emph{every even number
is the sum of two odd primes}. Goldbach's assertion
remains unproved; it has not even been proved that every
number~$n$ is the sum of $10$~primes, or of~$100$, or of any number
independent of~$n$. Our method is applicable in principle to
this problem also. We cannot solve the problem, but we
can open the first serious attack upon it, and bring it into
relation with the established prime number theory. The
most which we can accomplish at present is as follows.
We have to assume the truth of the notorious Riemann
hypothesis concerning the zeros of the Zeta-function, and
indeed in a generalized and extended form. If we do this
we can prove, not Goldbach's Theorem indeed, but the next
best theorem of the kind, viz.\ that \emph{every odd number}, at any
rate from a certain point onwards, \emph{is the sum of three odd
primes}. It is an imperfect and provisional result, but it is
the first serious contribution to the solution of the problem.
%% -----File: 037.png---
\BackMatter
\null
\vfill
\subsection*{\centering\normalfont{POSTSCRIPT}}
Srinivasa Ramanujan, F.R.S., Fellow of Trinity
College, Cambridge, died in India on April~26, 1920,
aged~32.

An account of his life and mathematical activities
will be published in Vol.~19 of the \textit{Proceedings of
the London Mathematical Society}.
\vfill
%% -----File: 038.png---
\newpage
\null
\vfill
\begin{center}
PRINTED IN ENGLAND \\
AT THE OXFORD UNIVERSITY PRESS \\
\vfill
\end{center}

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% End of the Project Gutenberg EBook of Some Famous Problems of the Theory of
% Numbers and in particular Waring's Problem, by G. H. (Godfrey Harold) Hardy
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK FAMOUS PROBLEMS OF THEORY OF NUMBERS ***
%                                                                         %
% ***** This file should be named 37030-t.tex or 37030-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/3/7/0/3/37030/                         %
%                                                                         %
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[]\OT1/cmtt/m/n/8 Title: Some Famous Problems of the Theory of Numbers and in p
articular Waring's Problem[] 
 []


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[]\OT1/cmtt/m/n/8 *** START OF THIS PROJECT GUTENBERG EBOOK FAMOUS PROBLEMS OF 
THEORY OF NUMBERS ***[] 
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[]\OT1/cmtt/m/n/8 *** END OF THIS PROJECT GUTENBERG EBOOK FAMOUS PROBLEMS OF TH
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    book.cls    2005/09/16 v1.4f Standard LaTeX document class
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inputenc.sty    2006/05/05 v1.1b Input encoding file
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amsfonts.sty    2001/10/25 v2.2f
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caption3.sty    2007/01/07 v3.0k caption3 kernel (AR)
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fancyhdr.sty    
geometry.sty    2002/07/08 v3.2 Page Geometry
geometry.cfg
hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
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hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
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     url.sty    2005/06/27  ver 3.2  Verb mode for urls, etc.
 hpdftex.def    2007/02/07 v6.75r Hyperref driver for pdfTeX
ragged2e.sty    2003/03/25 v2.04 ragged2e Package (MS)
everysel.sty    1999/06/08 v1.03 EverySelectfont Package (MS)
   color.sty    2005/11/14 v1.0j Standard LaTeX Color (DPC)
   color.cfg    2007/01/18 v1.5 color configuration of teTeX/TeXLive
  pdftex.def    2007/01/08 v0.04d Graphics/color for pdfTeX
supp-pdf.tex
 nameref.sty    2006/12/27 v2.28 Cross-referencing by name of section
refcount.sty    2006/02/20 v3.0 Data extraction from references (HO)
 37030-t.out
 37030-t.out
    umsa.fd    2002/01/19 v2.2g AMS font definitions
    umsb.fd    2002/01/19 v2.2g AMS font definitions
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Output written on 37030-t.pdf (46 pages, 247814 bytes).
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