398 History of the Theory of Numbers. (Chap, xvii 
By use of (1) and (2), theorems I-IV are proved. Theorem VIII is 
stated, and VII is proved. Employing two diagrams and working to base 2, 
he showed that 2"^^ — 1 is a prime. 
Lucas" considered a product m = p^i* ... of powers of primes, no one 
dividing Q. Set A = (a-6)^ (A/p)=0, ^\=^^r>-l)/2 ^^^^ ^^^ 
[51? 
,W=."-V-...[p-g)][.-(f)] 
Then w<=0 (mod m) for f =^(w). The ranks n of terms u^ divisible by m 
are multiples of a certain divisor /z o{\p{m). This ii is the exponent to which 
a or 6 belongs modulo vi. The case 6 = 1 gives Euler's generalization of 
Fermat's theorem. The primality test^^ is reproduced and applied to show 
that 2^^ — 1 is a prime. 
Lucas^^ considered the series of Pisano. Taking a, 6 = (l=tv5)/2, we 
have Ml =1*2 = 1, 1*3 = 2, etc. According as n is odd or even the divisors of 
u-iJun are divisors of 5x^ — 3?/^ or 5a:^+32/"; those of u^Ju2n are divisors of 
5x^ — 2zf or 5x^'-f-2?/^; those of v^Jvn are divisors of x^+Sy^ or x^ — 3y^; those 
of V2n are divisors of x^+22/^ or x^ — 2y^; those of u^JUn are divisors of x^+Sy^ 
or T' — hy^. The law V of repetition of primes and theorem III are stated. 
The law VIII of apparition of primes now takes the following form: If p is a 
' prime 10g=t 1, Wp_iis divisible by p ; if p is a prime lOg^ 3, i/p+i is divisible by p. 
The test^^ for the primality of A is given and applied to show that 2^^^ — 1 
and 2^^ — 1 are primes. There is a table of prime factors of u^ for n^60. 
L Finally, ^u^ju^ is expressible in the form x^ — 'py^ or bx^+py"^ according as 
the prime p is of the form 45-+! or 40^+3. 
Lucas^^ considered the series defined by r„+i = r„^ — 2, 
Let A = 3or9 (mod 10), g=0 (mod 4) ; or A = 7, 9 (mod 10), g=l( mod 4); 
or A=l, 7 (mod 10), g=2 (mod 4); or A=l, 3 (mod 10), q=S (mod 4). 
Then p = 2'A — 1 is a prime if the rank of the first term divisible by p is 5 ; 
if a {a<q) is the rank of the first term divisible by p, the divisors of p are 
either of the form* 2aAk-{-l, or of the forms of the divisors of x^ — 2y^ 
and x^ — 2Ay'^. Corresponding tests are given for 2^ A -\- 1 and S'^A — 1. The 
first part of the theorem of Pepin^'^ for testing the primaUty of a„ = 2^"+l 
follows from theorem VII with a = 5, 5 = 1, p = a„; the second part follows 
from the reciprocity theorem and the form of a„ — 1 . 
For A=p, let the above rj become r. When p=7 or 9 (mod 10) and 
p is a prime, then 2p — 1 is a prime if and only if r=0 (mod 2p — 1). When 
p = 4g+3 is a prime, 2pH-l is a prime if and only if 2''=1 (mod 2p + l). 
When p = 4^-4- 3 is a prime, 2p — 1 is a prime if and only if 
*'Comptes Rendus Paris, 84, 1877, 439-442. Corrected by Carmichael.*' 
"Bull. Bibl. Storia Sc. Mat. e Fis., 10, 1877, 129-170. Reprinted as " Recherches sur plusieura 
ouvrages de Leonard de Pise." Cf . von Sterneck" of Ch. XIX. 
"Assoc, frang. avanc. sc, 6, 1877, 1.59-166. *Corrected to 2MA'=tl in Lucas"; see Lucas." 
'"KUomptes Rendus Paris, 85, 1877, 329-331. See Ch. XV, Pepin", Lucas,"' " Proth.^^ 
