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G. H. HARDY :
p|n
and this is the formula to which Brun’s argument naturally
leads *). The formula, like Sylvester’s, is wrong, and by a
factor 4^~2C = i . 263 . . . . It will be observed that Sylvester’s
formula (7) is the geometric mean between (9) and (13).
I have explained that I do not believe that a proof of
Goldbach’s theorem is likely to be found by methods such as
these. But it is certainly possible to prove something in
this sort of way, and what Brun has proved seems to me
very interesting indeed. He has proved, for example that
every large even number n can be expressed in the
form m + m’, where m and m’ are numbers composed
of at most 9 odd prime factors, and that the number
of such decompositions is of order ;z:(log;z)2 at least*
His method of proof is elementary enough, but a little
complicated, and the idea which underlies it can probably be
explained most clearly by reference to a simpler problem.
The number of numbers not exceeding x, and prime or
not divisible by any of the primes /1? /2,- - -,/r, is (tf pr<^x)
A - AA
where [x] is the largest integer in x. This can be shown at
once by the method of Eratosthenes, and it is an imediate
deduction that
where A is a constant. No very interesting consequences can
be drawn from this; but Brun has shown that, if we are
content to allow the first term on the right of (15) to be multiplied
by a constant less than i, we can materially reduce the order
of the second, considering as a function of r. More precisely,
he proves that
The formulae (13) is not enunciated explicitly by Brun. His procedure
was rather to develop his argument until it leads to a formula of this
type, and to attempt to determine the constant on other grounds. The
determination of the only possible constant by averaging was effected
independently by Stäckel, and by Mr. Littlewood and myself.
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