30 History of the Theory of Numbers. [Chap, i 
{e. g., n = ll, 59, 83, 131, 179, 251). If n is a prime 24a:+23 and 2'*-l is 
composite, the least factor is of the form 48?/ +47 (e. g., n = 47, ?/ = 48, factor 
2351; n = 23, 71, 191, 239). Gerardin^^^ gave tables of the possible, but 
(unverified, factors of 2" — 1, n<257. 
A. Cunningham^^o gave the factor 150287 of 2^^^-\. 
A. Cunningham^^^ found the factor 228479 of 2^^-l. 
T. M. Putnam^^^ proved that not all of the r distinct prime factors of a 
perfect number exceed 1 +r/loge2 and hence do not all equal or exceed 1 +3r/2. 
L. E. Dickson^^^ gave an immediate proof that every even perfect num- 
ber is of Euclid's type. Let 2"g be perfect, where q is odd and n > 0. Then 
(2"+^ — l)s = 2'^'^^q, where s is the sum of all the divisors of q. Thus s = q-\-d, 
where d = q/{2''^^ — l). Hence d is an integral divisor of q, so that q and d 
are the only divisors of q. Hence d = \ and 5 is a prime. 
H. J. WoodalP^^ obtained the factor 43441 of 2^^^-l. 
R. E. Powers^^^ verified that 2^^ — 1 is a prime by use of Lucas' test on 
the series 4, 14, 194, .... H. Tarry^^^ made an incomplete examination. 
E. Fauquembergue^^^ proved that 2^^ — 1 is a prime by writing the residues 
of that series to base 2. 
A. Cunningham^^^ noted that 2^ — 1 is composite for three primes of 8 digits. 
On the proof-sheets of this history, he noted that the first two should be 
g = 67108493, p = 134216987; 5 = 67108913, p = 134217827. 
A G^rardin^^^'' observed that 2'^''+^-\=F^-2(?, F = 2"+^±l = 2m+l, 
G = 2"±l, G2 = m2+(m+l)2-(2")2. 
H. Tarry^^^^ verified for the known composite numbers 2^—1, where p 
is a prime, that, if a is the least factor, 2" — 1 is composite. 
A. Gerardin added empirically that, if p is any number and a any di- 
visor of 2^ — 1 , a = 8m =t 1 not being of the form 2" — 1 then 2" — 1 is composite. 
A. Cunningham^^^ noted that, if g is a prime, 
M^ = 2^-\ = T^-2{quY={qtf-2U\ 
If Mq is a prime it can be expressed in the forms A^-]-?>B^ — G'^-\-QH'^, and 
in one or the other of the pairs of forms f^au^ {ci = '^, 14, 21, 42). He 
discussed M^ to the base 2. 
>'»Sphinx-Oedipe, 3, 1908-9, 118-120, 161-5, 177-182; 4, 1909, 1-5, 158, 168; 1910, 149, 166. 
""Proc. London Math. Soc, (2), 6, 1908, p. xxii. 
"iL'intermddiaire des math., 16, 1909, 252; Sphinx-Oedipe, 4, 1909, 4e Trimestre. 36-7. 
"2Amer. Math. Monthly, 17, 1910, 167. ^^Ibid., 18, 1911, 109. 
'"Bull. Amer. Math. Soc, 16, 1910-11, 540 (July, 1911). Proc. London Math. Soc, (2), 9, 
1911, p. xvi. Mem. and Proc. Manchester Literary and Phil. Soc, 56, 1911-12, No. 1, 5 pp. 
Sphinx-Oedipe, 1911, 92. Verification by J. Hammond, Math. Quest. Solutions, 2, 
1916, 30-2. 
i«Bull. Amer. Math. Soc, 18, 1911-12, 162 (report of meeting Oct., 1911). Amer. Math. 
Monthly, 18, 1911, 195. Sphinx-Oedipe, Feb., 1912, 17-20. 
"«Sphinx-Oedipe, Dec, 1911, p. 192; 1912, 15. (Proc. London Math. Soc, (2), 10, 1912, 
Records of Meetings, 1911-12, p. ii.) 
^"Ibid., 1912, 20-22. "^Messenger Math., 41, 1911, 4. 
"saBuU. Soc. Philomatiquesde Paris, (10), 3, 1911, 221. isseSphinx-Oedipe, 6, 1911, 174 186, 192. 
""Math. Quest. Educ Times, (2), 19, 1911, 81-2; 20, 1911, 90-1, 105-6; 21, 1912, 58-9, 73. 
