390 History of the Theory of Numbers. [Chap, xvi 
A. Cunningham"^ discussed quasi-Mersenne numbers N'g = x^—y^, with 
x — y=l, q a prime, tabulating every prime factor < 1000 for q<50, a:<20 
if q>o, a:<50 if q = 5, and treated Aurifeuillians 
{X''=^Y')/{X=^Y), X = ^^, Y = qy]\ 
H. C. Pocklington"^ proved that, if n is prime, (x" — ?/'*)/(x — ?/) is divisible 
only by numbers of the form 77m + 1 unless x — y\s, divisible by n [Euler], 
and then is divisible only by n and numbers of the forms mn-\-\, n{inn-\-\). 
G. Fonten^*° stated that, if p is a prime and x, y are relatively prime, 
each prime factor of {x^ — y^)/{x — y) is of the form A'p + 1, except for a 
factor p, occurring if x=y (mod p) and then only to the first power if p>2. 
G. Fonten^^^ considered the homogeneous form/t(x, y) derived from (1) 
by setting a = x/y. If p" is the highest power of a prime p dividing n. 
The main theorem proved is the following: If x, y are relatively prime 
every prime divisor of /„(x, y) is of the form kn+l, 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 /„/pa=0 (mod p), the latter having for each y prime 
to p a number of roots x equal to the degree of the congruence. In par- 
ticular, if n is a power of a prime p, every prime factor of /„ is of the form 
kn-\-l, with the exception of a divisor p occurring if x=y (mod p), and then 
to the first power if n5^2. 
J. G. van der Corput^^ considered the properties of the factors of the 
expression derived from a' +6' as (1) is derived from a' — 1. 
A. G^rardin^ factored a^+6^ in four numerical cases and gave 
(a2+3iS2)H(4ai3)*=n{(3a2=fc2ai3+3/32)2-2(2a2±2a/3)2). 
A. Cunningham^ tabulated factors of y'^^2, 2?/*± 1. 
R. D. CarmichaeP^ treated at length the numerical factors of a"=tj8" 
and the homogeneous form Qnio-, /3) of (1), when a+^S and a/5 are relatively 
prime integers, while a, j8 may be irrational. 
A. G^rardin^S'^ factored xHl for x = 373, 404, 447, 508, 804, 929; inves- 
tigated x'* -2 for x^ 50, y^-Sfory^ 75, Sv^ - 1 for y^ 25, 2w^ -Iforw^ 37, 
and gave ten methods of factoring numbers Xa^ — 1. 
L. Valroff^^'' factored 2x^-1 for 101^x^180, 8x^-1 for x<128. 
A. Gerardin^^*^ expressed 622833161 (a factor of 20^°+ 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. Cambr. Phil. Soc, 16, 1911, 8. 
'"Nouv. Ann. Math., (4), 9, 1909, 384; proof, (4), 10, 1910, 475; 13, 1913, 383-4. 
"Ubid., (4), 12, 1912, 241-260. 
8«Nieuw Archicf 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. 
w^Sphinx-Oedipe, 1912, 188-9; 1913, 34-44; 1914, 20, 23-8, 34-7, 48. 
o^^lbid., 1914, 5-6, 18-9, 28-30, 33, 37, 73. 
*^Ibid., 39. Stated by E. Fauquembergue, I'interm^diaire des math., 21, 1914, 45. 
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