394 History of the Theory of Numbers. [Chap, xvn 
prime of which 6 is a non-residue, and having the form 2mn — 1, vdW di\'ide V. 
li b= —n, where n is a prime 4m-\-3, no prime divides V unless it is of the 
form A-n=fc 1, and conversely. The divisors of U are discussed for the case 
n a power of 2; in particular, of U = x^+Qbx^-\-b^ when n = 4. 
J. P. M. Binet^° noted that the number of terms of a solution t'„, expressed 
as a function of n, ro, . . ., of the equation t'„+2 = i'„+i+''«i'n in finite differ- 
ences is 
^(i^-'-(^)-*')- 
This equals f/„ as shown by taking each r„ to be unity. 
G. Lam6^^ used the series of Pisano^ to prove that the number of divi- 
sions necessary to find the g. c. d. of two integers by the usual process of 
division does not exceed five times the number of digits in the smaller 
integer. Lionnet^- added that the number of divisions does not exceed three 
times it when no remainder exceeds half the corresponding di\'isor. See 
also Serret, Traits d'Arithmetique; C. J. D. Hill, Acta Univ. Lundensis, 
2, 1865, No. 1; E. Lucas, Nouv. Corresp. Math., 2, 1876, 202, 214; 4, 
1878, 65, and Th^orie des Nombres, 1891, 335, Ex. 3; P. Bachmann, Niedere 
Zahlentheorie, 1902, 116-8; L. Grosschmid, Math.-Naturwiss. Blatter, 8, 
1911, 125-7, for an elementary proof by induction; Math, es Phys. Lapok, 
23, 1914, 5-9; R. D. Carmichael, Theory of Numbers, p. 24, Ex. 2. 
H. Siebeck^^ considered the recurring series defined by 
for a, c relatively prime. By induction, 
where /3 = or 1, 7 = (r — 1)/2 or (r — 2)/2, according as r is odd or even; 
whence A^^ is divisible by N^- If P and q are relatively prime, Np and 
A"g are relatively prime and conversely. If p is a prime, 6 = a"+4c, and 
s = (b/p) is Legendre's symbol, then 
Np=s, Np.,= (mod p), 
so that either Np+i or A"p_i is divisible by p. 
J. Dienger^^ considered the question of the number of terms of the series 
of Pisano with the same number of digits and the problem to find the rank 
of a given term. 
A. Genocchi^^ took a and b to be relatively prime integers and proved 
that B„„ is divisible by B„ and that the quotient Q has no odd divisor in 
"Comptea Rendus Paris, 17, 1843, 563. 
"Ibid., 19, 1S44, 867-9. Cf. Binet, pp. 937-9. 
'*Compl^ment des 616ments d'arithm^tique, 1857, 39-42. 
"Jour, fur Math., 33, 1846, 71-6. "Archiv Math. Phys., 16, 1851, 120-4. 
"AnnaU di Mat., (2), 2, 1868-9, 256-267. Cf. Genocchi". ". 
