396 HISTORY OP THE THEORY OF NUMBERS. [CHAP. XYII If the term of rank A +1 in Pisano's series is divisible by the odd number A of the form 10p=±= 3 and if no term whose rank is a divisor of A +1 is divisible by A , then A is a prime. If the term of rank A — 1 is divisible by A = lOp =*= 1 and if no term of rank a divisor of A — 1 is divisible by A, then A is a prime. It is stated that A = 2127— 1 is a prime since A = 10p— 3 and uk is never divisible by A f or k = 2n, except for n = 127. Lucas19 employed the roots a, b of a quadratic equation 3?— Px+Q=Q, where P, Q are relatively prime integers. Set n - _, w a — o The quotients of $un V — 1 and #n by 2Qn/2 are functions analogous to the sine and cosine. It is stated that t ... (1) (2) Not counting divisors of Q or S2, we have the theorems: (I) upq is divisible by up, us, and by their product if p, q are relatively prime. (II) uny vn are relatively prime. (III) If d is the g. c. d. of ra, n, then wd is the g. c. d of um, un. (IV) For n odd, un is a divisor of or2— Qy2. By developing unp and vnp in powers of un and vn, we get formulas analogous to those for sin nx and cos nx in terms of sin n and cos n, and thus get the law of apparition of primes in the recurring series of the un [stated explicitly in Lucas20], given by Fermat when 6 is rational and by Lagrange when 5 is irrational. The developments of unp and vnp as linear functions of ^n? ^2n> - . • are like the formulas of de Moivre and Bernoulli for smpx and cospx in terms of sin kx, cos kx. Thus — (V) If n is the rank of the first term un containing the prime factor p to the power \, then u^ is the first term divisible by px+1 and not by px4"2; this is called the law of repetition of primes in the recurring series of un. (VI) If p is a prime 4g+l or 4g+3, the divisors of upn/un are divisors of x2—py2 or 62x2+p2/2, respectively. (VII) If wp=hl is divisible by p} but no term of rank a divisor of p=*= 1 is divisible by p} then p is a prime, Lucas20 proved the theorems stated in the preceding paper. Theorems II and IV follow from (12) and (22), while (2^ shows that every factor common to um+n and um divides un and conversely. (VIII) If a and b are irrational, but real, up+i or up^.l is divisible by the prime p, according as 52 is a quadratic non-residue or residue of p (law of apparition of primes in the w's). If a and b are integers, wp_i is divisible by p. Hence the proper divisors of un are of the form kn-\-l if 5 is rational, kn =±=1 if d is irrational. 10Comptes Rendus Paris, 82, 1876, pp. 1303-5. 20Sur la th<§orie des nombres premiers, Atti R. Accad. Sc. Torino (Math.), 11, 1875-6, 92&-Q37.