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2 G. H. HARDY:
the sum of three. You will see later why this apparently
trivial remark should be interesting to me.
The numerical evidence for the truth of the theorem is
overwhelming. It has been verified up to 1000 by Cantor
(1894-1895), to 2000 by Aubry (1896 - 1903), to 10,000 by
Haussner (1896); and further numerical data, concerning special
numbers or numbers of specified forms, have been accumulated
by Ripert (1903), by Cunningham (1906), and by Shah and
Wilson (1919). But most of the modern computations have
been directed towards a more ambitious end, that of
determining or verifying some asymptotic formula for the number
of decompositions into primes of a given even number n.
I denote by v (ri) the number of ways in which n = 2N
can be expressed as a sum of two primes, or the number of
solutions of the equation
n - 2N = /+/’. (i)
According to Goldbach’s Theorem, v(«)^>o if ;z^>2; and a
very hasty survey of the evidence is enough to make two
things clear. In the first place,, a great deal more is true
than is asserted by the theorem. Not merely is v (ri) positive,
but it is large when n is large. Secondly, while v (n) tends
to infinity with n, it does not do so in any very regular
manner. The magnitude of v (ri) does not depend merely on
the magnitude of n, but also on its arithmetic form.
It is important to observe that these conclusions are in
complete agreement with the a priori judgement of common
sense. We naturally argue thus. If m <^ n, and n is large, the
chance that m is prime is approximately i : log n. If then
we write n in every possible way in the form n = m -\- m’,
the chance that both m and m’ are prime is approximately
i : (log n)2. We should therefore expect the order of magnitude
of v (ri) to be
On the other hand, the arithmetical form of n is plainly
also relevant. In the first place, it is obviously relevant
whether n is odd or even : if n is odd there is no representation,
unless n ~p -\- 2, and then only one. It is not quite so
obviously important to consider whether n is a multiple of 3.
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