424 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvin J. Merlin139 considered the operation A(b, 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} p^, Afa, pt-), A(r't-, pt-), t = 2, . . ., n, where pn is the nth prime >1, is equivalent to constructing a crib of Eratosthenes up to pn. It is stated that hi every interval of length vpn log pn 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. Vecchi140 wrote pn for the nth odd prime and called ph and ph+a of the same order if p2h>Ph+a* Then 2n>132 is a sum of two primes of the same order in [J W>+1)] ways if and only if there exist <j> numbers not >n— 2Vfi-f-l and not representable in any of the forms a,+3x, bi+Sx,..., li+pmx (x = l, 2), where pm+\ is the least prime p for which p2+p> 2n, and the known terms aif . . . 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. Goldbach145 stated empirically that every odd number is of the form p+2a2, where p is a prime and a is an integer ^ 0. L. Euler146 verified this up to 2500. Euler124 verified for m=8JV+3^187 that m is the sum of an odd square and the double of a prune 4n+l. J. L. Lagrange147 announced the empirical theorem that every prune 4n— 1 is a sum of a prime 4m +1 and the double of a prime 4/i-f-l. A. de Polignac148 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 later148" admitted to be in error for 959. M. A. Stern149 and his students found that 53409 = 5777 and 13-641 = 5993 are neither of the form p+2a2 and verified that up to 9000 there are no further exceptions to Goldbach's146 assertion. Also, 17, 137, 227, 977, 1187 and 1493 are the only primes <9000 not of the form p+2b2, 6>0. Thus all odd numbers <9000, which are not of the form 6n+5, are of the form p+262. E. Lemoine160 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 — 2ir ""Comptes Rendus Paris, 153, 1911, 516-8. Bull. des. sc. math., (2), 39, 1, 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. ^Ibid., p. 596, 606; Dec. 16, 1752. M7Nouv. Me"m. Ac. Berlin, anne"e 1775, 1777, 356; Oeuvres, 3, 795. "«Nouv. Ann. Math., 8, 1849, 428 (14, 1855, 118). ""Comptes Rendus Paris, 29, 1849, 400, 738-9. "»Nouv. Ann. Math., 15, 1856, 23. "°L'intermediaire des math., 1, 1894, 179; 3, 1896, 151