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GOLDBACH S THEOREM. 7
wrong in just the same way as Sylvester’s formula (7). Our
failure may help to deepen our scepticism as to the results to
be anticipated from the use of the principle of the sieve.
The ’sieve’ used by Merlin and Brun for Goldbach’s theorem
(le crible de Merlin) is of a slightly more complex kind.
We form the table
i, 2, 3, 4, 5, 6, 7, ...,;*- i (m\\
/nil1*)
n- I, 72-2, n- 3, n- 4, 72-5, «-ö, »- 7,–-, I (w ).j v
Here ;z is an even number, and the table, read vertically, shows
all possible decompositions n = m -\- m’ of n into positive
integers. We now perform the process of Eratosthenes on
both rows of the table, starting from the right of the lower
row, and operating with the first r primes. We consider a
decomposition m + wi to be erased when either of its
constituents is erased.
There are two cases to be considered for each prime p.
If / n, the erasures in the second row fall immediately below
the corresponding erasures in the first, and the number of
decompositions erased is approximately n:p. If /> + #, the
erasures never correspond, and their number is
approximately 2H : p. If then \T(n) is the number of decompositions into
numbers m, m1 which are prime or not divisible by any of
the first r primes, we have
n (-7) n
v-<">~" - – °2)
This formula is correct so long as r is fixed.
Let us assume once more that this formula is correct when
r is about /;z . We obtain
p|n
where the value 2 of/ is now excluded. But, if 3^/=S
we have
by Mertens’ formula (5). We thus obtain
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