CHAPTER XVI. 
FACTORS OF a^^b\ 
Fermat^ stated that (2''+l)/3 has no factors other than 2kp+l if p is 
an odd prime. 
L. Euler^ noted that o*+46* has the factors a^=i=2ah+2b^. 
Euler^ discussed the numbers a for which a^ + 1 is divisible by a prime 
4n+l=r^+s^. Let p/q be the convergent preceding r/s in the continued 
fraction for r/s ; then ps — qr==^l. Thus every a is of the form (4n + 1 ) m ± 
k, where k = pr+qs. 
Euler^ gave the 161 integers a< 1500 for which a^ + 1 is a prime, and the 
cases a = 1, 2, 4, 6, 16, 20, 24, 34 for which a*+l is a prime. 
Euler^ proved that, if m is a prime and a, b are relatively prime, a factor 
of a"* — 6"", not a divisor of a — 6, is of the form kn-\-l. If p = A;n + 1 is a prime 
and a =p=^ pa, then a*' — 1 is divisible by p. If of" — bg"" is divisible by a prime 
p = 'mn-{-l, while/ and g are not both divisible by p, then a"" — 6"" is divisible 
by p ; the converse is true if m and n are relatively prime. 
Euler^ proved the related theorems: For q an odd prime, any prime 
divisor of a^—1, not a divisor of a — 1, is of the form 2nq-\-l. If a"" — 1 is 
divisible by the prime p = m7i+l, we can find integers x, y not divisible by 
p such that A = ax'*— ?/" is divisible by p (since the quotient of a'"^""'— 2/"*" 
by A is not divisible by p\i x,y are suitably chosen) . 
Euler^ treated the problem to find all integers a for which a^ + 1 is divisible 
by a given prime 4n + 1 = p^ + ?^- If a^ + b^ is divisible by p'^-\-(f', there exist 
integers r, s such that a = pr+g's, 6 = ps—gr. We wish 6= =±=1. Hence we 
take the convergent r/s preceding p/q in the continued fraction for p/q. 
Thus ps — qr=^\, and our answer is a= ± {pr-\-qs). He listed all primes 
P = 4n+1<2000 expressed as p^+g^, and listed all the a's for which a^ + 1 
is divisible by P. The table may be used to find all the divisors < a of a 
given number a^+1. He gave his^ table and tabulated the values a< 1500 
for which {a'^-\-l)/k is a prime, for k=2, 5, 10. He tabulated all the 
divisors of a^+1 for a^ 1500. 
N. Beguelin^ stated that 2''+l has a trinary divisor 1+2^+2' only when 
n=10, 24, 32, although his examples (p. 249) contradict this statement. 
Euler^ gave a factor of 2"=^ 1 for various composite n's. 
iQeuvres, 2, 205, letter to Frenicle, Aug. (?), 1640. Bull. Bibl. St. Sc. Mat. e Fis., 12, 1879, 716. 
''Corresp. Math. Phys. (ed., Fuss), I, 1843, p. 145; letter to Goldbach, 1742. 
^lUd., 242-3; letter to Goldbach, July 9, 1743. 
mid., 588-9, Oct. 28, 1752. Published, Euler.^ 
^Novi Comm. Petrop., 1, 1747-8, 20; Coram. Arith. Coll., 1, 57-61, and posthumous paper, 
ibid., 2, 530-5; Opera postuma, I, 1862, 33-35. Cf. Euleri^z of Ch. VII and the topic 
Quadratic Residues in Vol. III. 
«Novi Comm. Petrop., 7, 1758-9 (1755), 49; Comm. Arith., 1, 269. 
^Novi Comm. Petrop., 9, 1762-3, 99; Comm. Arith., 1, 358-369. French transl., Sphinx- 
Oedipe, 8, 1913, 1-12, 21-26, 64. 
»M6m. Ac. Berlin, annee 1777, 1779, 255. Cf. Ch. XV and Henry.i^ 
'Posthumous paper, Comm. Arith., 2, 551; Opera postuma, I, 1862, 51. 
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