390 HlSTOEY OF THE THEORY OP NUMBERS. [CHAP. XVI A. Cunningham78 discussed quasi-Mersenne numbers Nq=xq--yq, with $— y sal, g a prime, tabulating every prime factor <1000 for #<50, cc<20 if <j>5, x < 50 if q=5j and treated Aurifeuillians H. C. Pocklington79 proved that, if n is prime, (xn—yn)/(x—y) is divisible only by numbers of the form ran+1 unless x— y is divisible by n [Euler], and then is divisible only by n and numbers of the forms mn+1, n(mn+l). G. Fontene*80 stated that, if p is a prune and x, y are relatively prune, each prime factor of (xp—yp)/(x—y) is of the form kp+1, except for a factor pj occurring if x^y (mod p) and then only to the first power if p>2. G. Fontene*81 considered the homogeneous form/«(a;, y) derived from (1) by setting a=x/y. If p* is the highest power of a prime p dividing n, /.- (/»/„.)*'•>, *n-r= W-fT (mod p). The main theorem proved is the following: If x, y are relatively prune every prune divisor of fn(x, y) is of the form kn+1, unless it is divisible by the greatest prime factor (say p) of n. It has this factor p if p — 1 is divisible by n/p* and if x, y satisfy /n/pa=0 (modp), the latter having for each y prime to p a number of roots x equal to the degree of the congruence. In particular, if n is a power of a prime p} every prime factor of fn is of the form kn+l9 with the exception of a divisor p occurring if x=y (mod p), and then to the first power if 71^2. J. G. van der Corput82 considered the properties of the factors of the expression derived from a?+V as (1) is derived from a'— 1. A. Ge'rardin83 factored os+68 in four numerical cases and gave (a2+3/32)4+ (4a/3)4 =n { (3a2=*= 2a/3+3/32)2 - A. Cunningham84 tabulated factors of 2/4=b 2, B. D. Carmichael85 treated at length the numerical factors of an=*=/3n and the homogeneous form Qn(a, /3) of (1), when a-f/ft and aft are relatively prime integers, while a, ft may be irrational. A. Ge"rardin85a factored z4+l for 3 = 373, 404, 447, 508, 804, 929; investigated z4-2 for a^50, ^-8 for j/^75, 8^-1 for v^25, 2w?4-l for w^37, and gave ten methods of factoring numbers Xa4 — 1. L. Valroff865 factored 2z4-l for 101 ^z^ 180, 8z4-l for z<128. A. G6rardin85c expressed 622833161 (a factor of 2010+1) as a sum of two squares in two ways to get its prime factors 2801 and 222361. "Messenger Math., 41, 1911-12, 119-145. "Proc. Gambr. Phil. Soc., 16, 1911, 8. 80Nouv. Ann. Math., (4), 9, 1909, 384; proof, (4), 10, 1910, 475; 13, 1913, 383-4. «IWd., (4), 12, 1912, 241-260. MNieuw Archie! voor Wiskunde, (2), 10, 1913, 357-361. "Wiskundig Tijdschrift, 10, 1913, 59. "Messenger Math., 43, 1913-4, 34-57. "Annals of Math., (2), 15, 1913-4, 30-70. «aSphinx-Oedipe, 1912, 188-9; 1913, 34-44; 1914, 20, 23-8, 34-7, 48. Kblbid., 1914, 5-6, 18-9, 28-30, 33, 37, 73. Htlbid., 39. Stated by E. Fauquembergue, 1'interme'diaire des math., 21, 1914, 45.