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406                   HISTORY OF THE THEORY OF NUMBERS.          [CHAP, xvii
R. Niewiadomski87 noted that, for a series of Pisano, ffJUsW*fi   or    -fflo-i
according as the prime N = 10m=*= 1 or 10w=*= 3. He showed how to compute rapidly distant terms of the series of Pisano and similar series, and factored numerous terms.
L. Bastien88 employed a prime p and integer ^<p and determined 2j 03,   -,each <p,by means of a^^Q, (modp).   Then
The types of series are found and enumerated. Every divisor of Kp is of the form Xp==l. Some of Lucas' results are given.
R. D. Carmichael89 generalized many of Lucas'38'39 theorems and corrected several. The following is a generalization (p. 46) of Fermat's theorem: If a+/3 and a/3 are integers and a/3 is prime to n=pil. . .pf*, where pi, . . ., pk are distinct primes, ux=(ax /3x)/(a /3) is divisible by n when X is the 1. c. m. of
(3)                         P<rMj>i-(,0W       -!,...,*).
Here, if p is an odd prime, the symbol (a, /3)p denotes 0, +1 or  1, according as (a /3)2 is divisible by p, is a quadratic residue of p, or is a quadratic non-residue of p'} while (a, 18)2 denotes +1 if a/3 is even, 0 if a/3 is odd and a+/3 is even, and  1 if a/3(a+/3) is odd. In particular, if <j> is the product of the numbers (3), i^=0(mod n), which is the corrected form of the theorem of Lucas'27.
Relations have been noted90 between terms of recurring series defined by one of the equations
E. Malo91 and Prompt92 considered the residues with respect to a prime modulus lOra^l of the series u0, HI, ^2 = ^0+^1, . . ., un = /wrt_i+wn_2-
A. Boutin93 noted relations between terms of Pisano's series.
A. Agronomof94 treated ^n = wni+wn_2H-wn_3.
Boutin95 and Malo95 treated sums of terms of Pisano's series.
A. Pellet96 generalized Lucas'28 law of apparition of primes.
A. G6rardin97 proved theorems on the divisors of terms of Pisano's series.
87L'interm6diaire des math., 20, 1913, 51, 53-6.
88Spliinx-Oedipe, 7, 1912, 33-38, 145-155.
"Annals of Math., (2), 15, 1913, 30-70.
Math. Quest. Educat. Times, 23, 1913, 55; 25, 1914, 89-91.
91I/interm<*diaire des math., 21, 1914, 8fr-8.
/Wd., 22, 1915, 31-6.                                         MMathesis, (4), 4, 1914, 125.
"Mathesis, (4), 4, 1914, 126.                               ^L'interm^diaire des math., 23, 1916, 42-3.
"I/intermediaire des math., 23, 1916, 64-7          7Nouv. Ann. Math., (4), 16, 1916, 361-7.