406 History of the Theory of Numbers. [Chap, xvii 
R. Niewiadomski^^ noted that, for a series of Pisano, 
Uff^a=Ula-hi or -Uia-1 (rnodN), 
according as the prime A^ = 10m±lorlOw±3. He showed how to compute 
rapidly distant terms of the series of Pisano and similar series, and factored 
numerous terms. 
L. Bastien^^ employed a prime p and integer a^Kp and determined 
02,03, . . .,each <p,bymeansof 0102= Q, 02+ 03—-^, 0304— Q,a4+a5=-P, • . . 
(mod p). Then 
a2.H-i^f^±^^^^-^'' (modp), K,+, = PK,-QK,_,. 
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. CarmichaeP^ generalized many of Lucas'^^'^^ theorems and 
corrected several. The following is a generalization (p. 46) of Fermat's 
theorem: If a-\-^ and aj3 are integers and aj3 is prime to n = pC\ . .pk'', 
where p\, . . .,Pk are distinct primes, Uy, = {a^—0^)/{a—^) is divisible by n 
when X is the 1. c. m. of 
(3) p;rMPi-(a,/3)pj {i=i,...,k). 
Here, if p is an odd prime, the symbol (a, ^)p denotes 0, +1 or — 1, according 
as {a—^Y is divisible by p, is a quadratic residue of p, or is a quadratic non- 
residue of p; while (a, /3)2 denotes +1 if a/3 is even, if a^ is odd and a-\-^ is 
even, and —1 if aj8(a+j3) is odd. In particular, if </> is the product of the 
numbers (3), •w^=0(mod n), which is the corrected form of the theorem of 
Lucas'". 
Relations have been noted®° between terms of recurring series defined by 
one of the equations 
^n + W„ + l='W„+2, 'W„ + Wn+2 = W„+3, y„+i + t'„_i = 4y„, ^1=1,^2 = 3. 
E. Malo^^ and Prompt^ ^ considered the residues with respect to a prime 
modulus 10m='=l of the series Uq, Ui, U2 = Uo-\-Ui,. . ., ii„ = w„_i+w„_2. 
A. Boutin^^ noted relations between terms of Pisano's series. 
A. Agronomof^^ treated it„ = it„_i+it„_2+w„_3. 
Boutin^^ and Malo^^ treated sums of terms of Pisano's series. 
A. Pellet^'' generalized Lucas' ^^ law of apparition of primes. 
A. G^rardin^^ proved theorems on the divisors of terms of Pisano's 
series. 
»'L'interm6diaire des math., 20, 1913, 51, 53-6. 
"Sphinx-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. 
"L'intermddiaire des math., 21, 1914, 86-8. 
"Ibid., 22, 1915, 31-6. "Mathesis, (4), 4, 1914, 125. 
"Mathesis, (4), 4, 1914, 126. "L'interm^diaire des math., 23, 1916, 42-3. 
••L'intermediaire des math., 23, 1916, 64-7 "Nouv. Ann. Math., (4), 16, 1916, 361-7. 
