400 History of the Theory of Numbers. [Chap, xvii 
Lucas^^ proved III by use of (2i) and gave 
Lucas^^ determined the quadratic forms of divisors of i'2n from 
t;2n=Aw„2+2Q", V2n = vJ'-2Q\ 
In the last, take Q = 2f, n = 2/1+1; thus 1^4^+2 factors if Q is the double of a 
square. As a special case we have the result by H. LeLasseur (p. 86) : 
In the first expression for t'2n, take n=juH-l, A = ±2/i^, Q==r^ff^; thus 
Vi^+2 factors when QA is of the form — 2f. Similarly, v^^ factors if A = —2f. 
Lucas" gave the formulas 
developments of w„^, t'/ as linear functions of Vkn, k = p, p — 2, p — i,. . ., and 
complicated developments of w„r, ^nr- 
Lucas^^ reproduced the preceding series of seven papers, added (p. 228) 
a theorem on the expression of 4Upr/ur as a quadratic form, a proof (p. 231) 
of his^^ test for primality by use of the s^, and results on primes and perfect 
numbers cited elsewhere. 
Lucas^^ considered series w„ of the first kind (in which the roots a, h are 
relatively prime integers) and deduced Fermat's theorem and the analogue 
it<=0 (mod m), t = 4>(m), of Euler's generalization. Proof is given of the 
earlier theorems VII, VIII and (p. 300) of his" generalization of the Euler- 
Fermat theorem. The primality test^^ is stated (p. 305) and applied to 
show that 2^^ — 1 and 2^^ — 1 are primes. It is stated (page 309) that 
p = 2'**+^ — 1 is prime if and only if 
3=2coS7r/22«+i (mod p), 
after rationalizing with respect to the radicals in the value of the cosine. 
The primality tests^^ are given (page 310), with similar ones for 3'yl + l, 
2-5'A + l. The tests^^ for the primaUty of 2p + l are given (p. 314). The 
primality test^^ for 2'^'''^^ — 1 is proved (pp. 315-6). 
Lucas"*" reproduced his^^ earlier results, and for p = 3, 5, 7, 11, 13, 17, 
expressed ypr/i'2r in the form x^— 2pQV> and, for p a prime ^31, expressed 
»»Nouv. Corresp. Math., 4, 1878, 65-71. 
"Ihid., pp. 97-102. 
*Ubid., pp. 129-134, 225-8. 
»«Amer. Jour. Math., 1, 1878, 184-220. Errors noted by Carmichael." 
"Ibid., pp. 289-321. 
"Atti R. Accad. Sc. Torino, 13, 1877-8, 271-284. 
