Chap. XVII] RECURRING SERIES ; LuCAS' Un, V^. 397 
The law V of repetition of primes follows from 
where t = (p — l)/2. Special cases of the law are due to Arndt,^^ p. 260, 
and Sancery,^^ each quoted in Ch. VII. Theorem VII, which follows from 
VIII, gives a test for the primality of 2"± 1 which rests on the success of 
the operation, whereas Euler's test for 2^^ — 1 was based on the failure of 
the operation. The work to prove that 2^^ — 1 is prime is given, and it is 
stated that 2^'' — 1 was tested and found composite, ^^ contrary to Mersenne. 
Finally, a^+Qy^ is shown to have an infinitude of prime divisors. 
A. Genocchi^^ noted that Lucas' w„, y„ are analogous to his^^ 5„, A^. 
[If we set a = a+\/b, ^ = a — \/h, we have 
a — p 
Lucas^^ stated that, if 4m+3 is prime, p = 2*'"+^ — 1 is prime if the first 
term of the series 3, 7, 47,. . ., defined by r„_,.i = r„^ — 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 2"A;=t:l, together with the divisors of 
x^ — 2y^. There are analogous tests by recurring series for the primality of 
3.24m+3_i^ 2-3*™+2±i^ 2-3*'"+^-l, 2-52'"+' + l. 
Lucas^^ 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 w„+2 
= aUn+i-\-bUn', also the proof of VIII: If p is a prime, 
(a-\-Vby-^-(a-Vby-'^ 
Up^l—- 
Vb 
is divisible by p if 6 is a quadratic residue of p, excepting values of a for 
which a^ — 6 is divisible by p; and the corresponding result [of Lagrange^ and 
Gauss^] for Up+i. Moret-Blanc^^ gave a proof by use of the binomial theorem 
and omission of multiples of p. 
Lucas^^ wrote s„ for the sum of the nth powers of the roots of an 
equation whose coefficients are integers, the leading one being unity. 
Then Snp — sJ' is an integral multiple of p. Take n = 1. Then Si = impHes 
Sp= (mod p) . It is stated that if Si = and if Sj, is divisible by p for k = p, 
but not for k<p, then p is a prime. 
*iA. Cunningham, Proc. Lond. Math. Soc, 27, 1895-6, 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. 
22Atti. R. Accad. Sc. Torino, 11, 1875-6, 924. 
^Comptes Rendus Paris, 83, 1876, 1286-8. 
2*Nouv. Ann. Math., (2), 15, 1876, 82. 
^mid., (2), 20, 1881, 258 [p. 263, for primality of 2"-l]. 
28Assoc. frang. avanc. sc, 5, 1876, 61-67. Cf. Lucas^*. 
