Generalized Lucas primitive part

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

Definitions and Notes

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))).

Record Primes of this Type

rank prime digits who when comment 1 primV(111534, 1, 27000) 72683 x25 Nov 2013 Generalized Lucas primitive part 2 primV(27655, 1, 19926) 57566 x25 Mar 2013 Generalized Lucas primitive part 3 primV(40395, - 1, 15588) 47759 x23 Feb 2007 Generalized Lucas primitive part 4 primV(53394, - 1, 15264) 47200 CH4 May 2007 Generalized Lucas primitive part 5 primV(4836, 1, 16704) 39616 x25 Jan 2013 Generalized Lucas primitive part 6 primV(38513, - 1, 11502) 34668 x23 Nov 2006 Generalized Lucas primitive part 7 primV(9008, 1, 16200) 34168 x23 Nov 2005 Generalized Lucas primitive part 8 primV(6586, 1, 16200) 32993 x25 Jan 2013 Generalized Lucas primitive part 9 primV(28875, 1, 13500) 32116 x25 Jul 2016 Generalized Lucas primitive part 10 primV(10987, 1, 14400) 31034 x25 Aug 2005 Generalized Lucas primitive part 11 primV(24127, - 1, 6718) 29433 CH3 Oct 2005 Generalized Lucas primitive part 12 primV(12215, - 1, 13500) 29426 x25 Jul 2016 Generalized Lucas primitive part 13 primV(45922, 1, 11520) 28644 x25 Apr 2011 Generalized Lucas primitive part 14 primV(5673, 1, 13500) 27028 CH3 Sep 2005 Generalized Lucas primitive part 15 primV(44368, 1, 9504) 26768 CH3 Sep 2005 Generalized Lucas primitive part 16 primV(10986, - 1, 9756) 26185 x23 Dec 2005 Generalized Lucas primitive part 17 primV(11076, - 1, 12000) 25885 x25 Nov 2005 Generalized Lucas primitive part 18 primV(17505, 1, 11250) 25459 x25 Apr 2011 Generalized Lucas primitive part 19 primV(42, - 1, 23376) 25249 x23 Sep 2007 Generalized Lucas primitive part 20 primV(7577, - 1, 10692) 25140 x33 Apr 2007 Generalized Lucas primitive part

References

Carmichael1913

R. D. Carmichael, "On the numerical factors of the arithmetic forms αⁿ ± βⁿ," Ann. Math., 15 (1913) 30--70.

Lucas1878

E. Lucas, "Theorie des fonctions numeriques simplement periodiques," Amer. J. Math., 1 (1878) 184--240 and 289--231.

Morrison75

M. Morrison, "A note on primality testing using Lucas sequences," Math. Comp., 29 (1975) 181--182.  MR 51:5469

Ribenboim95

P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]

Schinzel1974

A. Schinzel, "Primitive divisors of the expression Aⁿ - Bⁿ in algebraic number fields," J. Reine Angew. Math., 268/269 (1974) 27--33.  MR 49:8961

Stewart1977

C. L. Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers," Proc. Lond. Math. Soc., 35:3 (1977) 425--447.  MR 58:10694

Voutier1995

Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences," Math. Comp., 64:210 (1995) 869--888.  MR1284673 (Annotation available)

Ward1959

M. Ward, "Tests for primality based on Sylvester's cyclotomic numbers," Pacific J. Math., 9 (1959) 1269--1272.  MR 21:7180