Lehmer primitive part

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Definitions and Notes

Lehmer defined a generalization of Lucas sequences as follows:

where a and b are the zeros of the polynomial z²-R^1/2z+Q for non-zero rational integers R, Q, and R-4Q.

A primitive divisor of a Lehmer is one that does not divide any previous term, and does not divide the product R(R-4Q). Many of the references below (culminating in [BHV2002]) show that all but a finite number of the terms in a Lehmer sequence have primitive divisors.

If we let n=3^rm^s, where m > 3 is prime and min(r,s)>0. Then the numbers

(V(P,1,n/3)+1)/(V(P,1,n/(3m)+1)
(V(P,1,n/3)-1)/(V(P,1,n/(3m)-1)

are Lehmer primitive parts, with R=P+2 and Q=1, and their product is the Lucas primitive part primU(P,1,n).

Record Primes of this Type

rank prime digits who when comment

1 (U(9275, 1, 3961)+U(9275, 1, 3960))/(U(9275, 1, 45)+U(9275, 1, 44)) 15537 x38 May 2009 Lehmer primitive part
2 (V(49596, 1, 3375)+1)/(V(49596, 1, 675)+1) 12678 x25 Jan 2006 Lehmer primitive part
3 (V(47025, 1, 3375)-1)/(V(47025, 1, 675)-1) 12616 x25 Jan 2006 Lehmer primitive part
4 (V(44524, 1, 3375)-1)/(V(44524, 1, 675)-1) 12552 x23 Jan 2006 Lehmer primitive part
5 (V(37511, 1, 3375)-1)/(V(37511, 1, 675)-1) 12351 x25 Jan 2006 Lehmer primitive part
6 (V(32362, 1, 3375)+1)/(V(32362, 1, 675)+1) 12178 x23 Jan 2006 Lehmer primitive part
7 (V(30226, 1, 3375)-1)/(V(30226, 1, 675)-1) 12098 x25 Jan 2006 Lehmer primitive part
8 (V(11258, 1, 3375)+1)/(V(11258, 1, 675)+1) 10939 x23 Jan 2006 Lehmer primitive part
9 (V(10638, 1, 3375)-1)/(V(10638, 1, 675)-1) 10873 x25 Jan 2006 Lehmer primitive part
10 (V(983, 1, 3609)-1)/(V(983, 1, 9)-1) 10774 x23 May 2006 Lehmer primitive part
11 (V(9352, 1, 3375)+1)/(V(9352, 1, 675)+1) 10722 x25 Dec 2005 Lehmer primitive part
12 (V(7792, 1, 3375)-1)/(V(7792, 1, 675)-1) 10508 x25 Jan 2006 Lehmer primitive part
13 (V(812, 1, 3609)+1)/(V(812, 1, 9)+1) 10475 x25 Apr 2006 Lehmer primitive part
14 (U(11987, 1, 2521)-U(11987, 1, 2520))/(U(11987, 1, 36)-U(11987, 1, 35)) 10136 x38 May 2009 Lehmer primitive part
15 (U(10227, 1, 2521)+U(10227, 1, 2520))/(U(10227, 1, 36)+U(10227, 1, 35)) 9965 x38 May 2009 Lehmer primitive part
16 (V(8259, 1, 2517)-1)/(V(8259, 1, 3)-1) 9848 x25 Dec 2005 Lehmer primitive part
17 (U(6954, 1, 2521)-U(6954, 1, 2520))/(U(6954, 1, 36)-U(6954, 1, 35)) 9548 x38 May 2009 Lehmer primitive part
18 (V(2247, 1, 3375)-1)/(V(2247, 1, 675)-1) 9050 x25 Jan 2006 Lehmer primitive part
19 (V(3798, 1, 2529)-1)/(V(3798, 1, 9)-1) 9021 x25 Feb 2006 Lehmer primitive part
20 (V(8162, 1, 2307)-1)/(V(8162, 1, 3)-1) 9013 x25 Dec 2005 Lehmer primitive part

References

BHV2002
Bilu, Yu., Hanrot, G. and Voutier, P. M., "Existence of primitive divisors of Lucas and Lehmer numbers," J. Reine Angew. Math., 539 (2001) 75--122. With an appendix by M. Mignotte. MR1863855 [From the review: "This remarkable paper answers completely a one century old problem, by proving that, for any integer n>30, the n-th element of any Lucas or Lehmer sequence has a primitive divisor."]
Schinzel1963
Schinzel, A., "On primitive prime factors of Lehmer numbers. II," Acta. Arith., 8 (1962/1963) 251--257. MR 27:1409
Schinzel1968
Schinzel, A., "On primitive prime factors of Lehmer numbers. III," Acta Arith., 15 (1968) 49--70. MR0232744
Schinzel1970
Schinzel, A., "Corrigendum to the papers "On two theorems of Gelfond and some of their applications" and "On primitive prime factors of Lehmer numbers. III"," Acta Arith., 16 (1969/1970) 101. MR0246840
Stewart1976
Stewart, C. L., Primitive divisors of Lucas and Lehmer numbers. In "Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976)," Academic Press, London, 1977. pp. 79--92, MR0476628
Stewart1977b
Stewart, C. L., Primitive divisors of Lucas and Lehmer numbers. In "Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976)," Academic Press, London, 1977. pp. 79--92, MR 57:16187
Stewart1983
Stewart, C. L., "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. III," J. London Math. Soc. (2), 28:2 (1983) 211--217. MR 85g:11021
Voutier1995
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences," Math. Comp., 64:210 (1995) 869--888. MR1284673 (Annotation available)
Voutier1996
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences. II," J. Th\'eor. Nombres Bordeaux, 8:2 (1996) 251--274. MR1438469
Voutier1998
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences. III," Math. Proc. Cambridge Philos. Soc., 123:3 (1998) 407--419. MR1607969 [From the review: "The main result of this paper is that for any integer n>30 030, the nth element of any Lucas or Lehmer sequence has a primitive divisor."]