CHAP, xvii] RECURRING SERIES; LUCAS* un, vn. 397
The law V of repetition of primes follows from
where t=(p — 1)/2. Special cases of the law are due to Arndt,33 p. 260, and Sancery,61 each quoted in Ch. VII. Theorem VII, which follows from VIII, gives a test for the primality of 2n± 1 which rests on the success of the operation, whereas Euler's test for 231 — 1 was based on the failure of the operation. The work to prove that 231~1 is prime is given, and it is stated that 267 — l was tested and found composite,21 contrary to Mersenne. Finally, a^+Qy2 is shown to have an infinitude of prime divisors.
A. Genocchi22 noted that Lucas' unj vn are analogous to his16 Bn, An. [If we set a = a+V5j /3 = a— V?, we have
Lucas23 stated that, if 4m +3 is prime, j)==24m+3~ 1 is prime if the first term of the series 3, 7, 47, . . ., defined by rn+1— rn2— 2, which is divisible by p is of rank 4m+2; but p is composite if no one of the first 4m +2 terms is divisible by p. Finally, if a is the rank of the first term divisible by p, the divisors of p are of the form 2tt&=t=l, together with the divisors of s2— 2?/2. There are analogous tests by recurring series for the primality of
3.24m-h3_lj 2-34m+2=tl, 2-34m+3-l, 2-52m+1-fl.
Lucas24 proposed as an exercise the determination of the last digit in the general term of the series of Pisano and for the series defined by un+2 = aun+i+bun; also the proof of VIII: If p is a prime,
""-1 -v/6
is divisible by p if 6 is a quadratic residue of p} excepting values of a for which a2 —6 is divisible by p; and the corresponding result [of Lagrange6 and Gauss8] for UP+I . Moret-Blanc25 gave a proof by use of the binomial theorem and omission of multiples of p.
Lucas26 wrote sn for the sum of the nth powers of the roots of an equation whose coefficients are integers, the leading one being unity. Then snp—snp is an integral multiple of p. Take n = 1. Then sx = 0 implies sp== 0 (mod p). It is stated that if 5X = 0 and if sk is divisible by p for k = p, but not for k<p, then p is a prime.
21A. Cunningham, Proc. Lond. Math. Soc., 27, l%95-6, 54, remarked that, while primality is proved by Lucas' process by the success of the procedure, his verification that a number is composite is indirect and proved by the failure of the process and hence is liable to error.
"Atti. R. Accad. Sc. Torino, 11, 1875-6, 924.
23Comptes Rendus Paris, 83, 1876, 1286-8.
MNouv. Ann. Math., (2), 15, 1876, 82.
»/Wd., (2), 20, 1881, 258 [p. 263, for primality of 2»-l].
MAssoc. franc., avanc. sc., 5, 1876, 61-67. Cf. Lucas38.