422 History of the Theory of Numbers. [Chap, xviii 
A. Desboves^^^ verified that every even number between 2 and 10000 is a 
sum of two primes in at least two ways; while, if the even number is the 
double of an odd number, it is simultaneously a sum of two piimes of the 
form 4n+l and also a sum of two primes of the form 4n — 1. 
J. J. Sylvester^ ^^ stated that the number of ways of expressing a very 
large even number n as a sum of two primes is approximately the ratio of the 
square of the number of primes < n to n, and hence bears a finite ratio to the 
quotient of n by the square of the natural logarithm of n. [Cf. Stackel"^]. 
F. J. E. Lionnet^-' designated by x the number of ways 2a can be 
expressed as a sum of two odd primes, by y the number of ways 2a can be 
expressed as a sum of two distinct odd composite numbers, by z the number 
of odd primes <2a, and by q the largest integer ^a/2. He proved that 
q^x = y-\-z and argued that it is very probable that there are values of n 
for which q = y-\-z, whence a: = 0. 
N. V. Bougaief ^^^" noted that, if M{n) denotes the number of ways n can 
be expressed as a simi of two primes, and if Oi denotes the ith. prime >1, 
S(n-3^.)ilf(n-^.)=0. 
G. Cantor^^^ verified Goldbach's theorem up to 1000. His table gives 
the number of decompositions of each even number < 1000 as a sum of two 
primes and lists the smaller prime. 
V. Aubry^29 verified the theorem from 1002 to 2000. 
R. Haussner^^° verified the law up to 10000 and announced results 
observed by a study of his^" tables up to 5000. His table I (pp. 25-178) 
gives the number v of decompositions of every even n up to 3000 as a sum 
x-\-y oi two primes and the values oi x (x^y), as in the table by Cantor. 
His table II (pp. 181-191) gives v for 2<n<5000; this table and further 
computations enable him to state that Goldbach's theorem is true for 
n< 10000. Let P(2p+1) be the number of all odd primes 1, 3, 5, . . . which 
are ^2p+l, and set 
^(2p+l)=P(2p+l)-2P(2p-l)+P(2p-3), P(-l)=P(-3)=0. 
Then the number of decompositions of 2n into a sum of two primes x, y 
{x^y) is „_i 
S P(2n-2p-l)^(2p+l). 
If e = 1 or — 1 according as n is a prime or not, 
»/ = iSP(2n-2p-l)^(2p+l)+^- 
■1 
2 
i»Nouv. Ann. Math., 14, 1855, 293. 
i"Proc. London Math. Soc, 4, 1871-3, 4-6; CoU. M. Papers, 2, 709-711. 
>"Nouv. Ann. Math., (2), 18, 1879, 356. Cf. Assoc, frang. av. ec, 1894, I, p. 96. 
""KDomptes Rendus Paris, 100, 1885, 1124. 
"'Assoc, fran?. av. sc, 1894, 117-134; rinterm^diaire des math., 2, 1895, 179. 
"»L'interm6diaire des math., 3, 1896, 75; 4, 1897, 60; 10, 1903, 61 (errata, p. 166, p. 283). 
"•Jahresbericht Deutschen Math.-Verein., 5, 1896, 62-66. Verhandlungen Gesell. Deutscher 
Naturforscher u. Aerzte, 1896, II, 8. 
»"Nova Acta Acad. Caes. Leop.-Carolinae, 72, 1899, 1-214. 
