Chap. XVIII] GoLDBACh's THEOREM. 423 
Table III gives the values of P and ^ for each odd number 2p+l<5000. 
P. StackeP^^ noted that Lionnet's^" argument is not conclusive, and 
designated by G2r, the number of all decompositions of 2n as a sum of two 
primes (counting p+g and g+p as two different decompositions). If 
Pk is the number of all odd primes from 1 to k, 
I G2y'^ = {l^x^f = {l-x^Y(%P,^^,:^''A\ 
where p ranges over all the odd primes. Approximations to G^2n for n large 
in terms of Euler's 0-function are 
[P(2n- V2n) -P{-V2^)f n 
2n 
0(2n) n-y/2n <t>{2n) 
where P{k) is written for P^ for convenience in printing. Lack of agree- 
ment with Sylvester ^2^ is noted; cf. Landau. ^^^ It is stated that the 
truth of Goldbach's theorem is made very probable [but not proved^^^]. 
Sylvester^^^" stated that any even integer 2n is a sum of two primes, one 
> n/2 and the other < 3n/2, whence it is possible to find two primes whose 
difference is less than any given number and whose sum is twice that number. 
F. J. Studnicka^^* discussed Sylvester's statement. 
Sylvester^^**" stated that, if N is even and X, . . . , co are the 6 primes > \N 
and <fA^ (excluding ^A^" if it be prime), the number of ways of composing 
N [by addition] with two of these primes is the coefficient of x^ in 
(l^+-+T^)Vr(.-l)r- (rfe2). 
E. Landau^^^ noted that Stackel's approximation to G„ is 
mn= 
n^ 
log^ n0(n) 
and showed that S^=i(t„ has the true approximation ^x^/\o^x. By a longer 
analysis, he proved that if we use Stackel's (SJ„ to form the sum, we do not 
obtain a result of the correct order of magnitude. 
L. Ripert^^^ examined certain large even numbers. 
E. Maillet^" proved that every even number ^350000 (or 10^ or 9-10®) 
is, in default by at most 6 (or 8 or 14), the sum of two primes. 
A. Cunningham^^^ verified Goldbach's theorem for all numbers up to 
200 million which are of the forms 
(4-3)", (4-5)", 2•10^ 2'»(2"=f1), a-2", 2a", (2a)^ 2(2^=Fa), 
for a = 1, 3, 5, 7, 9, 11. He reduced the formula of Haussner for i' to a form 
more convenient for computation. 
"2G6ttingen Nachrichten, 1896, 292-9. "'Encyclop6die des sc. math., I, 17, p. 339, top. 
i33aNature, 55, 1896-7, 196, 269. »"Casopis, Prag, 26, 1897, 207-8. 
"40Educ. Times, Jan. 1897. Proof by J. Hammond, Math. Quest. Educ. Times, 26, 1914, 100. 
"^Gottingen Nachrichten, 1900, 177-186. 
"«L'interm6diaire des math., 10, 1903, 67, 74, 166 (errors, p. 168). 
»"/6id., 12, 1905, 107-9. »"Messenger Math., 36, 1906, 17-30. 
^ 
