Chap. XVII] RECURRING SERIES ; LuCAS' W„, V^. 399 
:^{a+V2y-a-V2r}^o (mod2p-i). 
To_test the primality of p = 2^^^^ — l, use x^ — 4x-{-l=0 with the roots 
2± Vs. Then if p is a prime, Wp+i is divisible by p. We use the residues 
of the series 2, 7, 97, . . . defined by r„+i = 2r„^ — 1. 
Lucas^^ stated that p = 2'^'""*"^ — 1 is a prime if the rank of the first term^^ 
of 3, 7, 47, . . . divisible by p is between 2m and 4m +2. To test P = 2^«+^ - 1, 
form the series 
ri = l, r2=-l, rs=-7, r, = 17,..., r„+i = 2r„2-32"-'; 
if ^ is the least integer for which r^ is divisible by P, then P is a prime when I 
is comprised between 2g and 4g+l, composite when ?>4g' + l. 
Lucas^^ expressed Un, v^ as polynomials in P and A = P^ — 4Q = 5^, obtained 
various relations between them corresponding to relations between sine and 
cosine; in particular, 
Wn+2 — P'^n+l ~ Q'^ni '^n+2r — ^r'^n+r ~ Q ^n> 
and formulas derived from them by replacing uhy v; also symbolic formulas 
generalizing those^^ for the series of Pisano. 
In the second paper, Un+i, v^ are expressed as determinants of order n 
whose elements are Q, — P, 2, 1, 0. There is given a continued fraction for 
U(n+i)r/unr, hoTCi whlch Is derfved (I2) and generalizations. The same 
fraction is developed into a series of fractions. 
Lucas^^ noted that u^r is divisible by Ur since 
where ^ = Jn — 1 if n is even, f = J(n — 1) if n is odd, the final factor being then 
absent. Proof is given for (2i) and 2y^+„ = ?;^y„+AM„w^. From these are 
derived new formulas by changing the sign of n and applying 
To show that 
[m, n\ = — 
is integral, apply (2i) repeatedly to get 
2[m, n\ = [m — l, n]Vn+{m, n — \]v^. 
Finally, sums of squares of functions Un, v^ are found. 
Lucas^^ gave a table of the linear forms 4A+?' of the odd divisors of 
x^-\-Llf and x^—I\'if' for A = l,. . ., 30. By use of (I2), it is shown that the 
terms of odd rank in the series u^ are divisors of x^ — Qif' ; the terms of even or 
odd rank in the series v^ are divisors of x^+A?/^ or x^-\-QAy^, respectively. 
3iMessenger Math., 7, 1877-8, 186. 
'^Sur la theorie des fonctions numeriques simplement p^riodiques, Nouv. Corresp. Math., 3, 
1877, 369-376, 401-7. These and the following five papers were reproduced by Lucas.'* 
^Hhid., 4, 1878, 1-8, continuation of preceding. 
^'lUd., pp. 33-40. 
