f 
424 History of the Theory of Numbers. [Chap, xviii 
J. Merlin^^' considered the operation A (6, a) of effacing from the natural 
series of integers all the numbers ax-\-b. The effect of carrying out one of 
the two sets of operations A{ri, pi), ^(r,-, p,), A{r'i, pi), i = 2,. . ., n, where 
p„ is the nth prime > 1, is equivalent to constructing a crib of Eratosthenes 
up to p„. It is stated that in every interval of length vp^ log p„ there is at 
least one number not effaced, if v is independent of n. It is said to follow 
that, for a sufficiently large, there exist two primes having the sum 2a. 
Under specified assumptions, there exist an infinitude of n's for which 
Pn+i-Pn = 2. 
M. Vecchi^*° wrote Pn for the nth odd prime and called p^ and p/,+„ of 
the same order if p^h>Ph+a- Then 2n> 132 is a sum of two primes of the 
same order in [K0+1)] ways if and only if there exist </> numbers not 
> /I — p^+i + 1 and not representable in any of the forms 
Gi+Sx, bi+5x,..., li+PmX (i=l, 2), 
where p^+i is the least prime p for which p'^-\-p> 2n, and the known terms a., 
. . . are the residues with respect to the odd prime occurring as coefficient of x. 
*G. Giovannelli, Sul teorema di Goldbach, Atri, 1913. 
Theorems Analogous to Goldbach's. 
Chr. Goldbach ^^^ stated empirically that every odd number is of the 
form p-\-2a^, where p is a prime and a is an integer ^ 0. L. Euler^^^ verified 
this up to 2500. Euler^24 verified for m = 8iV+3^ 187 that m is the sum 
of an odd square and the double of a prime 4n+l. 
J. L. Lagrange ^"^^ announced the empirical theorem that every prime 
4n — 1 is a sum of a prime 4?n + l and the double of a prime 4/i + l. 
A. de Polignac^^^ conjectured that every even number is the difference 
of two consecutive primes in an infinitude of ways. His verification up 
to 3 million that every odd number is the sum of a prime and a power of 
2 was later "^^ admitted to be in error for 959. 
M. A. Stern^^^ and his students found that 53-109 = 5777 and 13-641 
= 5993 are neither of the form p-\-2a^ and verified that up to 9000 there are 
no further exceptions to Goldbach's^'*^ assertion. Also, 17, 137, 227, 977, 
1187 and 1493 are the only primes <9000 not of the form p+26^ 6>0. 
Thus all odd numbers <9000, which are not of the form 6n+5, are of the 
form p+26^. 
E. Lemoine^^'' stated empirically that every odd number >3 is a sum 
of a prime p and the double of a prime tt, and is also of the forms p — 2Tr 
and 27r' — p'. 
'"Comptes Rendus Paris, 153, 1911, 516-8. Bull. des. sc. math., (2), 39, I, 1915, 121-136. In 
a prefatory note, J. Hadamard noted that, while the proof has a lacuna, it is suggestive. 
""Atti Reale Accad. Lincei, Rendiconti, (5), 22, II, 1913, 654-9. 
'"Corresp. Math. Phys. (ed.. Fuss), 1, 1843, 595; letter to Euler, Nov. 18, 1752. 
»«/W<i., p. 596, 606; Dec. 16, 1752. 
"'Nouv. M6ni. Ac. Berlin, ann6e 1775, 1777, 356; Oeuvres, 3, 795. 
"8Nouv. Ann. Math., 8, 1849, 428 (14, 1855, 118). 
»«»''Comptes Rendus Paris, 29, 1849, 400, 738-9. 
"»Nouv. Ann. Math., 15, 1856, 23. ""L'interm^diaire des math., 1, 1894, 179; 3, 1896, 151 
