History Of The Theory Of Numbers - I

CHAP, xvii] RECTJBKING SEKIBS; LUCAS' uny vn. 399
-'0 (xnod2p-l).
Tojest the primality of p = 24<rfl-l, use «2~4a;+l = 0 with the roots 2± V3. Then if p is a prime, wp+1 is divisible by p. We use the residues of the series 2, 7, 97, ... defined by rn+1=2rn2-l.
Lucas31 stated that p = 24m+3 — 1 is a prime if the rank of the first term23 of 3, 7, 47, ... divisible by pis between 2m and 4m +2. To test P=24fl+1~l, form the series
ri-1, ra--l, r3=-7, r4=17,..., rn+1-2rn2~32n^;
if I is the least integer for which rz is divisible by P, then P is a prime when I is comprised between 2q and 4#+l, composite when Z>4#+1.
Lucas32 expressed un, vn as polynomials in P and A = P2 — 4Q =S2, obtained various relations between them corresponding to relations between sine and cosine; in particular,
Un+2 = PWn+1 ~ Qu>n, Un+2r = VTUn+r - QrUn,
and formulas derived from them by replacing u by v\ also symbolic formulas generalizing those16 for the series of Pisano.
In the second paper, wn+1, vn are expressed as determinants of order n whose elements are Q, •— P, 2, 1, 0. There is given a continued fraction for tt(n+1)r/Mrtr, from which is derived (12) and generalizations. The same fraction is developed into a series of fractions.
Lucas33 noted that unr is divisible by ur since
where i = \ n — 1 if n is even, t • = \(n — 1) if n is odd, the final factor being then absent. Proof is given for (2^ and 2,vm+n = vmvn+&unum. From these are derived new formulas by changing the sign of n and applying
W_n==
To show that
is integral, apply (2X) repeatedly to get
2[m, n] = [m — 1, n]vn+[m, n — l]vm.
Finally, sums of squares of functions un) vn are found.
Lucas34 gave a table of the linear forms 4A+r of the odd divisors of a^+A?/2 and x2—Ay2 for A = 1, . . ., 30. By use of (12), it is shown that the terms of odd rank in the series un are divisors of x2 — Qy2; the terms of even or odd rank in the series vn are divisors of x2+A2/2 or o^+QAy2, respectively.
"Messenger Math., 7, 1877-8, 186.
32Sur la th^orie dee fonctions numeriques simplement peYiodiques, Nouv. Corresp. Math., 3, 1877, 369-376, 401-7. These and the following five papers were reproduced by Lucas.38 33/Wd., 4, 1878, 1-8, continuation of preceding. M/Wd.f pp. 33-40.