400 HlSTOKY OF THE THEOBT OF NUMBEES. [CHAP. XVII Lucas86 proved III by use of (2^ and gave ^^ Lucas36 determined the quadratic forms of divisors of v2n from In the last, take Q = 2g2, n=2/i+l; thus t^+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 =/*+!, A==fc2/i2, Q==F02; thus 1^+2 factors when QA is of the form 21? . Similarly, v^ factors if A = 2J2. Lucas37 gave the formulas developments of unp, vn* as linear functions of vkn, k=p, p2, p~4,..., and complicated developments of unrt vnr. Lucas38 reproduced the preceding series of seven papers, added (p. 228) a theorem on the expression of 4uw/ur as a quadratic form, a proof (p. 231) of his26 test for primality by use of the $t, and results on primes and perfect numbers cited elsewhere. Lucas39 considered series u* of the first kind (in which the roots a, 6 are relatively prime integers) and deduced Fermat's theorem and the analogue UfS^O (mod m)j £=<£(m), of Euler's generalization. Proof is given of the earlier theorems VII, VIII and (p. 300) of his27 generalization of the Euler-Fermat theorem. The primality test23 is stated (p. 305) and applied to show that 231 1 and 219 1 are primes. It is stated (page 309) that p==24fl+3 1 is prime if and only if 3=2 cos ir/22fl+1 (mod p), after rationalizing with respect to the radicals in the value of the cosine. The primality tests29 are given (page 310), with similar ones for 34A + 1, 2-5M.+L The tests29 for the primality of 2p+l are given (p. 314). The primality test29 for 2*9+l-l is proved (pp. 315-6). Lucas40 reproduced his36 earlier results, and for p = 3, 5, 7, 11, 13, 17, expressed vpr/v2r in the form a?2pQry2, and, for p a prime g31, expressed «Nouv. Corresp. Math., 4, 1878, 65-71. *'Ibid., pp. 97-102. 37Ibid., pp. 129-134, 225-8. "Amer. Jour. Math., 1, 1878, 184r-220. Errors noted by Carmichael.59 "Ibid., pp. 28&-321.