32 HlSTOBY OF THE THEOKY OF NUMBERS. [CHAP. I doubt: lfv 2 = 101> 103> 107> 109> 137> 139> 149> 157> f67> 193> 199> 227> 229 241 257- The last has no factor under one million, as verified by R E. Powers.1940 No one of the other 14 has a factor under one million, as verified twice with the collaboration of A. Ge"rardin. Up to the present three errors have been found in Mersenne's assertion; M67-has been proved composite (Lucas,116 Cole173), while M63. and M89 have been proved prime (PervusW40 Seelhoff,141 Cole,173 Powers185). It is here announced that M173 has the factor 730753, found with the collaboration of A. Ge*rardin. J. McDonnell195 commented on a test by Lucas hi 1878 for the primality of 2n-l. L. E. Dickson196 gave a table of the even abundant numbers <6232. R. Niewiadomski197 noted that 2761 1 has the factor 4567 and gave known factors of 2* 1. He gave the formula G Ricalde198 gave relations between the prunes p, q and least solutions of R. E. Powers199 proved that 2107 1 is a prime by means of Lucas'31 test in Ch. XVII. E. Fauquembergue200 proved that 2P 1 is prime for p = 107 and 127, composite for p** 101, 103, 109. T. E. Mason201 described a mechanical device for applying Lucas'118 method for testing the primality of 24fl+3 1. R. E. Powers202 proved that 2103 1 and 2109-1 are composite by means of Lucas' tests with 3, 7, 47, . . .and 4, 14, 194. .. (Ch. XVII), respectively. A. Ge"rardin203 gave a history of perfect numbers and noted that 2P 1 can be factored if we find i such that m~2pt+l is a prune not dividing s=l+2p+ 22p+ . . . +2(2<-1)p, since 22pr-ls(2p-l)« (mod ra). Or we may seek to express 23) 1 in two ways in the form x*2y2. On tables of exponents to which 2 belongs, see Ch. VII, Cunningham and Woodall128, Kraitchik.125 ADDITIONAL PAPERS OF A MEKELY EXPOSITORY CHARACTER. E. Catalan, Mathesis, (1), 6, 1886, 100-1, 178. W. W. Rouse Ball, Messenger Math., 21, 1891-2, 34-40, 121. Font^s (on Bovillus20), M&n. Ac. Sc. Toulouse, (9), 6, 1894, 155-67. J. Bezdicek, Casopis Mat. a Fys., Prag, 25, 1896, 221-9. Hultsch (on lamblichus), Nachr. Kgl. Sachs. GeselL, 1895-6. H. Schubert, Math. Mussestunden, I, Leipzig, 1900, 100-5. M. Nassd, Revue de math. (Peano), 7, 1900-1, 52-53. 1MOSphinx-Oedipe, 1913, 49-50. 198London Math. Soc., Records of Meeting, Dec., 1912, v-vi. 198Quart. Jour. Math., 44, 1913, 274-7. 197L'interme*diaire dea math., 20, 1913, 78, 167. 19876id., 7-8, 149-150; cf. 140-1. l»Proc. London Math. Soc., (2), 13, 1914, Records of meetings, xxxix. Bull. Amer. Math. Soc., 20, 1913-4, 531. Sphinx-Oedipe, 1914, 103-8. 100Sphinx-Oedipe, June, 1914, 85,' rmterme*diaire des math., 24, 1917, 33. 201Proc. Indiana Acad. Science, 1914, 429-431. 2MProc. London Math. Soc., (2), 15, 1916, Records of meetings, Feb. 10, 1916, xxii. 2MSphinx-Oedipe, 1909, 1-26.