396 History of the Theory of Numbers. [Chap, xvii 
If the term of rank ^ + 1 in Pisano's series is divisible by the odd number A 
of the form 10^=*= 3 and if no term whose rank is a divisor of A + 1 is divisible 
by .4 , then A is a prime. If the term of rank A — 1 is divisible by A = 10p=t 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=2^^^ — 1 is a prime since A = 10p — 3 and u^ is never 
divisible by A for k = 2", except for n = 127. 
Lucas^^ employed the roots a, 6 of a quadratic equation x^— Px+Q = 0, 
where P, Q are- relatively prime integers. Set 
a" — 6" 
Un = r-' Vn = a"+h'*, 5 = o — 6. 
a — o 
The quotients of 5if„ V — 1 and y„ by 2^"^^ are functions analogous to the 
sine and cosine. It is stated that 
(1) U2n = UnVn, V^-bW = '^Q', 
(2) 2u^+ri = UmVn + U„Vm, "J^n^ " Itn-l^^n+l = Q'*"^ 
Not counting divisors of Q or 5^, we have the theorems: 
(I) Wpg is divisible by Up, u^, and by their product if p, q are relatively 
prime. 
(II) Un, Vn are relatively prime. 
(III) If d is the g. c. d. of m, n, then Ud is the g. c. d of u^, Un. 
(IV) For n odd, u^, is a divisor of x^ — Qif'. 
By developing w„p and f„p in powers of Un and v„, we get formulas analo- 
gous 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 it„ [stated 
explicitly in Lucas"*^], given by Fermat when 5 is rational and by Lagrange 
when 5 is irrational. The developments of uj' and v^^ as linear functions of 
""nj '^2n, ■ ■ ■ are like the formulas of de Moivre and Bernoulli for sin^x and 
cos^x in terms of sin kx, cos kx. Thus — 
(V) If n is the rank of the first term u^ containing the prime factor p 
to the power X, then Uj^ is the first term divisible by p^"*"^ and not by p^"*"^; 
this is called the law of repetition of primes in the recurring series of w„. 
(VI) If p is a prime 4g+l or 4g+3, the divisors of u^Ju^ are divisors 
of x^—py^ or S^x^+pi/^, respectively. 
(VII) If Up^i is divisible by p, but no term of rank a divisor of p=*=l is 
divisible by p, then p is a prime. 
Lucas^*^ proved the theorems stated in the preceding paper. Theorems 
II and IV follow from (I2) and (22), while (20 shows that every factor 
common to w^^.„ and u^ divides Un and conversely. 
(VIII) If a and h are irrational, but real, t^p+i or Wp_i is divisible by the 
prime p, according as 6^ is a quadratic non-residue or residue of p (law of 
apparition of primes in the it's). If a and h are integers, Up^i is divisible 
by p. Hence the proper divisors of u^ are of the form /en +1 if 6 is rational, 
/cn=*=l if 5 is irrational. 
"Comptes Rendus Paris, 82, 1876, pp. 1303-5. 
"Sur la thdorie des nombres premiers, Atti R. Accad. Sc. Torino (Math.), 11, 1875-6, 928-937. 
