# Full text of "History Of The Theory Of Numbers - I"

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```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.```