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30 HISTORY OF THE THEORY OF NUMBERS. [CHAP, i (e. g., n = ll, 59, 83, 131, 179, 251). If n is a prime 24z+23 and 2n-l is composite, the least factor is of the form 4%+47 (e. g., n=47, y = 48, factor 2351; n=23, 71, 191, 239). Ge'rardin179 gave tables of the possible, but (unverified, factors of 2n 1, n<257. A. Cunningham180 gave the factor 150287 of 2163- 1. A. Cunningham181 found the factor 228479 of 271-1. T. M. Putnam182 proved that not all of the r distinct prime factors of a perfect number exceed 1 +r/logfi2 and hence do not all equal or exceed 1 +3r/2. L. E. Dickson183 gave an immediate proof that every even perfect number is of Euclid's type. Let 2nq be perfect, where q is odd and n > 0. Then (2n+1 l)s = 2n+lq, where s is the sum of all the divisors of *q. Thus s = q+d, where d-q/(2n+1 1). Hence d is an integral divisor of q, so that q and d are the only divisors of q. Hence d= 1 and q is a prime. H. J. Woodall184 obtained the factor 43441 of 2181-1. R. E. Powers185 verified that 289 1 is a prime by use of Lucas' test on the series 4, 14, 194, ____ H, Tarry186 made an incomplete examination. E. Fauquembergue187 proved that 289 1 is a prime by writing the residues of that series to base 2. A. Cunningham188 noted that 25 1 is composite for three prunes of 8 digits. On the proof-sheets of this history, he noted that the first two should be 5 = 67108493, p = 134216987; q = 67108913, p = 134217827. A Ge>ardin188a observed that 22n+1-l=JP2-2G2, 7/T=2n+1=t=l=2m+l, H. Tarry188i verified for the known composite numbers 2P 1, where p is a prime, that, if a is the least factor, 2a 1 is composite. A. Gerardin added empirically that, if p is any number and a any divisor of 2P 1, a = 8w =*= 1 not being of the form 2U 1 then 2° 1 is composite. A. Cunningham189 noted that, if q is a prime, If Mq is a prime it can be expressed in the forms A2+3jB2 = (r2+6#2, and in one or the other of the pairs of forms J2=±=cm2 (a = 7, 14, 21, 42). He discussed Mq to the base 2. 17»Sphinx-Oedipe, 3, 1908-9, 118-120, 161-5, 177-182; 4, 1909, 1-5, 158, 168; 1910, 149, 166. 180Proc. London Math. Soc., (2), 6, 1908, p. xxii. 181LJinterm6diaire des math., 16, 1909, 252; Sphinx-Oedipe, 4, 1909, 4e Trimestre, 36-7. 18JAmer. Math. Monthly, 17, 1910, 167. 183/6id., 18, 1911, 109. 1MBull. 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. 188BuU. Amer. Math. Soc., 18, 1911-12, 162 (repprt of meeting Oct., 1911). Amer. Math. Monthly, 18, 1911, 195. Sphinx-Oedipe, Feb., 1912, 17-20. l86Sphinx-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. l880Bull. Soc. Philomatiquesde Paris, (10), 3, 1911,221. 1886Sphinx-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.