Chap. XVI] FACTORS OF a" =±=5". 387 
A. Cunningham^^ factored 5'* — ! for n = 75, 105. 
L. Kronecker^^" proved that every divisor, prime to <, of (1) is = 1 (mod t). 
H. S. Vandiver^®'' noted that the proof applies to the homogeneous form 
Ft{a, b) of (1) if a, b are relatively prime. 
D. Biddle^^ gave a defective proof that 3-2^^+1 is a prime. 
The Math. Quest. Educational Times contains the factorizations of: 
Vol. 66 (1897), p. 97, 2i55_i factor 3P. Vol. 68 (1898), p. 27, p. 112, 272o_i. 
p. 114, 1012+4. 
Vol. 69 (1898), p. 61, 3824+1; p. 73, x^-1, x = 500, 2000; p. 117, x'+y^; p. 118, 
10^^+33, 33.IOI8+I. 
Vol. 70 (1899), p. 32, p. 69, 242i''+l; p. 47, 3201^-1; p. 64, 2^2+1, 8i*+l, 
20018+1; p. 72, 2014-1; p. 107, 9721^+1. Vol. 71 (1899), p. 63, x^^+^-l;p. 72, 
x'-\-y\ 
Vol. 72 (1900), p. 61, (Sny^-l factor 24n+l if prime; p. 86, 72210+1; p. 117, 
144010+1. 
Vol. 73 (1900), p. 51, 3520+I; p. 96, 711-I; p. 104, p. 114, x'-\-y*. 
Vol. 74 (1901), p. 27, a prime 2iq+l divides g^-1 if k = 2^-^; p. 86, rcio-5y. 
Vol. 75 (1901), p. 37, 3^+y'; p. 90, 1792^+1; p. Ill, 7^5+1. [Educ. Times, 
(2), 54, 1901, 223, 260]. 
Ser. 2, Vol. 1 (1902), p. 46, 10082«+1; p. 84, x'-\-fxy^. Vol. 2 (1902), p. 33, p. 
53, iV4+l; p. 118, IP^+l. 
Vol. 3 (1903), p. 49, a'+b* (cf. 74, 1901, 44); p. 114, a«+l, a = 60000. 
Vol. 6 (1904), p. 62. 9618+1. 
Vol. 7 (1905), p. 62, 20813-1; pp. 106-7, 2126+I. 
Vol. 8 (1905), p. 50, 9618+1; p. 64, 212^+1. 
Vol. 10 (1906), p. 36, 5418+I, 6^4+1. 
Vol. 12 (1907), p. 54, 6*2+1, 24^0+1. 
Vol. 13 (1908), p. 63, 106-7, S'*-\-2'\ 
Vol. 14 (1908), p. 17, 15018+1; p. 71, sextics; p. 96, 7^5+1. 
Vol. 15 (1909), p. 57, S''-\-2'*; p. 33, 3111+I, 12*5+1; p. 103, 282i+l, 44ii+l, 
630+1. 
Vol. 16 (1909), p. 21, 1924+1. 
Vol. 18 (1910), pp. 53-5, 102-3, x^-\-^y'; pp. 69-71, a:«+27?/«; p. 93, y^^-l. 
Vol. 19 (1911), p. 103, c(^-\-y' = z^-\-w\ Vol. 23 (1913), p. 92, (x'^-Nx-^NY 
-{-Nix^-Ny. 
Vol. 24 (1913), pp. 61-2, x'^^^y^ y = 5, 7, 11, 13; pp. 71-2, a;i2+2«, x'^-\-S\ 
x3''+3i5. 
Vol. 26 (1914), p. 23, x^^-{-l for fc = 6n+35^3^ p. 39, a;i2+6«; p. 42, xl0-5^ 
^14+7^ a:22+llii, a;26-13i3. Vol. 27 (1915), pp. 65-6, 451^-1, 20''-l, fc^o+l for A; 
= 6, 8, 10; p. 83, x4+4y4 (when four factors). Vol. 28 (1915), p. 72, 503o+l. Vol. 
29(1916), p. 95, 9618+1. 
New series, vol. 1 (1916), p. 86, rc2o+10i<', x28+14i4; pp. 94-5, x^'^-5^^, x^'^+WK 
Vol. 2 (1916), p. 19, ajso-sis. 
Vol. 3 (1917), p. 16, x''-y''; p. 52, xH-l. 
E. B. Escott^^ gave many cases when 1+x^ is a product of two powers 
of primes or the double of such a product. 
«Proc. London Math. Soc, 34, 1901, 49. 
^^''Vorlesungeu iiber Zahlentheorie, 1, 1901, 440-1. 
s6*Amer. Math. Monthly, 10, 1903, 171. 
"Messenger Math., 31, 1901-2, 116 (error); 33, 1903-4, 126. 
"L'mterm^diaire des math., 7, 1900, 170. 
