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CHAP, xviii]                     GOLDBACH'S THEOREM.                                423
Table III gives the values of P and £ for each odd number 2p+l<5000. P. Stackel132 noted that Lionnet's127 argument is not conclusive, and designated by G2n the number of all decompositions of 2n as a sum of two primes (counting p+q and q+p as two different decompositions). If Pk is the number of all odd primes from 1 to k,
where p ranges over all the odd primes.   Approximations to G2n for n large in terms of Euler's ^-function are
where P(&) is written for Pk for convenience in printing. Lack of agreement with Sylvester126 is noted; cf. Landau.135 It is stated that the truth of Goldbach's theorem is made very probable [but not proved133].
Sylvester1330 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. Studnicka134 discussed Sylvester's statement.
Sylvester134" stated that, if N is even and X, . . . , co are the 0 primes > %N and < JAT (excluding JJV if it be prime), the number of ways of composing N [by addition] with two of these primes is the coefficient of XN in
E. Landau135 noted that Stackers approximation to Gn is
and showed that SJ»i(?w has the true approximation |x2/log2o?. By a longer analysis, he proved that .if we use Stackers &n to form the sum, we do not obtain a result of the correct order of magnitude.
L. Ripert136 examined certain large even numbers.
E. Msillet137 proved that every even number ^350000 (or 106 or 9-106) is, in default by at most 6 (or 8 or 14), the sum of two primes.
A. Cunningham138 verified Goldbach's theorem for all numbers up to 200 million which are of the forms
(4-3)n,    (4-5)n,   2-10n,    2n(2n=Fl),   a-2",   2an,    (2a)n,    2(2n=r=a),
for a = 1, 3, 5, 7, 9, 11. He reduced the formula of Haussner for v to a form more convenient for computation.
132G6ttingen Nachrichten, 1896, 292-9.            "'Encyclop&iie dee sc. math., I, 17, p. 339, top.
U30Nature, 55, 1896-7, 196, 269.                     "*Casopis, Frag, 26, 1897, 207-8.
134aEduc. Times, Jan. 1897.   Proof by J. Hammond, Math. Quest. Educ, Times, 26, 1914, 100.
"•Gdttingen Nachrichten, 1900, 177-186.
1ML'interme'diaire des math., 10, 1903, 67, 74, 166 (errors, p. 168).
™Ibid., 12, 1905, 107-9.                                  »8Messenger Math., 36, 1906, 17-30.