32 History of the Theory of Numbers. [Chap. I 
doubt: M„ q = 101, 103, 107, 109, 137, 139, 149, 157, 167, 193, 199, 227, 
229, 241, 257. The last has no factor under one million, as verified by 
R. E. Powers. ^^^^ No one of the other 14 has a factor under one milhon, as 
verified t^dce with the collaboration of A. G^rardin. Up to the present 
three errors have been found in Mersenne's assertion; Mqj has been proved 
composite (Lucas,^^° Cole^^^), while Mqi and Mgg have been proved prime 
(Pervusin,"° Seelhoff,^*^ Cole,^"^ Powers^^^). It is here announced that M^^ 
has the factor 730753, found with the collaboration of A. Gerardin. 
J. ]McDonnell^^^ commented on a test by Lucas in 1878 for the primality 
of 2"-l. 
L. E. Dickson^^® gave a table of the even abundant numbers <6232. 
R. Niewiadomski^^^ noted that 2^^^ — 1 has the factor 4567 and gave 
known factors of 2'* — 1. He gave the formula 
2^'"+^-l = (2^'"+2"*-l)^+(22"'-2'"-l)^ + l. 
G. Ricalde^^^ gave relations between the primes p, q and least solutions of 
22«+i_i = pg, al-2h^ = p, c'-2dr = q. 
R. E. Powers^^^ proved that 2^°" — 1 is a prime by means of Lucas'^^ test 
in Ch. XVII. 
E. Fauquembergue^'''' proved that 2^ — 1 is prime for p = 107 and 127, 
composite for p = 101, 103, 109. 
T. E. Mason-°^ described a mechanical de\'ice for applying Lucas'"^ 
method for testing the primality of 2^^'*'^ — 1. 
R. E. Powers^°^ proved that 2^°^ — 1 and 2^°^ — 1 are composite by means 
of Lucas' tests with 3, 7, 47, . . .and 4, 14, 194. . . (Ch. XVII), respectively. 
A. Gerardin^°^ gave a history of perfect numbers and noted that 2^—1 
can be factored if we find t such that m = 2pt-\-\ is a prime not dividing 
8 = 1+2^+22^+ . . . +2^2'-^^^ since 2-p'-1= (2^-1)8 (mod m). Or we may 
seek to express 2^—1 in two ways in the form x^—2y'. 
On tables of exponents to which 2 belongs, see Ch. VII, Cunningham 
and Woodam'^ Kraitchik.^-^ 
Additional Papers of a Merely Expository Character. 
E. Catalan, Mathesis, (1), 6, 1886, 100-1, 178. 
W. W. Rouse Ball, Messenger Math., 21, 1891-2, 34-40, 121. 
- Pontes (on Bovnius^"), Mem. 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. Nasso, Revue de math. (Peano), 7, 1900-1, 52-53. 
i»*'Sphinx-Oedipe, 1913, 49-50. 
i«*London Math. Soc, Records of Meeting, Dec, 1912, v-vi. 
"«Quart. Jour. Math., 44, 1913, 274-7. 
'"L'interm^diaire des math., 20, 1913, 78, 167. 
"s/btd., 7-8, 149-150; cf. 140-1. 
'"Proc. London Math. Soc, (2), 13, 1914, Records of meetings, xxxi.x. Bull. Amer. Math. 
Soc, 20, 1913^, 531. Sphinx-Oedipe, 1914, 103-8. 
20<>Sphinx-Oedipe, June, 1914, 85; I'interm^diaire des math., 24, 1917, 33. 
-"Proc Indiana Acad. Science, 1914, 429-431. 
2«Proc London Math. Soc, (2), 15, 1916, Records of meetings, Feb. 10, 1916, xxii. 
»"Sphinx-Oedipe, 1909, 1-26. 
