The Top Twenty: Generalized Lucas primitive part

To define the Lucas sequences, let a and b be the zeros of the polynomial x²-Px+Q (where P, Q and D = P²-4Q are non-zero integers), then define the two companion sequences as follows:

U_n(P,Q) = (aⁿ - bⁿ)/(a - b), and V_n(P,Q) = aⁿ + bⁿ.

These sequences are both called Lucas sequences, and the numbers in them are the generalized Lucas numbers.

Because of the way the Lucas sequences are defined, it is clear that if n and k are positive integers, then U_n divides U_kn (e.g., U_2n=U_nV_n). Similarly if n and k are positive integers with k odd, then V_n divides V_kn. This means we can write members of these sequences as products of previous terms of the sequence (those with subscripts dividing n) times a primitive part using the Möbius function.

The role of Lucas sequences in primality proving was begun in [Lucas1878] and cemented by [Morrison75]. Their primitive parts (also known as Sylvester's cyclotomic numbers) were studied in [Ward1959]. Prime generalized Lucas numbers are clearly a particular case of prime primitive parts, occurring when n is also a prime. As Ribenboim indicates, there is an extensive literature on primitive prime Lucas factors, from [Carmichael1913] to [Voutier1995], via, for example, [Schinzel1974] and [Stewart1977].

Note: As with many such forms, when the parameters are unrestricted, all primes are of these forms. So in keeping with our definition of generalized repunit primes we require that 5n > max(abs(p),abs(sqrt(D))).

rank

prime

digits

who

when

comment

primV(27655, 1, 19926)

57566

x25

Mar 2013

Generalized Lucas primitive part

primV(40395, - 1, 15588)