CHAP, xvii] EECUBRING SEKIBS; LUCAS' un, vn. Uv/Ur in the form Aa^pQV. The prime factors of 329=*=1 are given on p: 280. The proper divisors of 24n+l are known to be of the form Snq + 1 ; it is shown that q is even. Thus for 232+l the first divisor to be tried is 641, for 22l2+l the first one is 114689; hi each case the division is exact (cf. Ch. XV). The following is a generalization: If the product of two relatively prime integers a and 6 is of the form 4/H-l, the proper divisors of a2o&n+62a6n are of the form Sdbnq+1. A primality test for 24Q+3 — I is given. Finally, p=24nfl+2n"H-l is a prime if and only if (2n+V25^)^V(2n-V^^)E?=0 (mod p). T. Pepin41 gave a test for the primality of <j=2n— 1. Let 2(a2~b2) , , " Ul~ Wb2 (modg) and form the series HI, u2, . . . , un__i by use of wa+1=wa2— 2 (modq). Then q is a prime if and only if un^ is divisible by q. This test differs from that by Lucas23 in the choice of %. E. Lucas42 reproduced his29 test for the primality of 2qA — 1, etc., and the test at the end of another paper,40 with similar tests for 24<H"3 — 1 and 212<H~5 — 1 . G. de Longchamps43 noted that, if dk~uk— auk^i, with the generalization X II dp,=d,, Take PI= . . . =px = P- Hence There is a corresponding theorem for the t>'s. J. J. Sylvester44 considered the g. c. d. of ux, ux+i if ux = (2x - I E. Gelin46 stated and E.Cesdro46 proved by use of Un+p= UJ that, in the series of Pisano, the product of the means of four consecutive terms differs from the product of the extremes by =*=!; the fourth power of the middle term of five consecutive terms differs from the product of the other four terms by unity. "Comptes Rendus Paris, 86, 1878, 307-310. «Bull. Bibl. Storia Sc. Mat. e Fis., 1 1, 1878, 783-798. The further results are cited in Ch . XVI . Comptes Rendus, 90, 1880, 855-6, reprinted in Sphinx-Oedipe, 5, 1910, 60-1. "Nouv. Corresp. Math., 4, 1878, 85; errata, p. 128. "Comptes Rendus Paris, 88, 1879, 1297; Coll. Papers, 3, 252. «Nouv. Corresp. Math., 6, 1880, 384. "Ibid., 42S-4.