 # 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-R1/2z+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=3rms, 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

rankprime digitswhowhencomment
1(V(77786, 1, 6453) + 1)/(V(77786, 1, 27) + 1) 31429 x25 Dec 2012 Lehmer primitive part
2(V(73570, 1, 6309) - 1)/(V(73570, 1, 9) - 1) 30661 x25 Jun 2016 Lehmer primitive part
3(V(28286, 1, 6309) + 1)/(V(28286, 1, 9) + 1) 28045 x25 Jun 2016 Lehmer primitive part
4(V(59936, 1, 4863) + 1)/(V(59936, 1, 3) + 1) 23220 x25 Jan 2013 Lehmer primitive part
5(V(45366, 1, 4857) + 1)/(V(45366, 1, 3) + 1) 22604 x25 Feb 2013 Lehmer primitive part
6(V(23354, 1, 4869) - 1)/(V(23354, 1, 9) - 1) 21231 x25 Feb 2013 Lehmer primitive part
7(V(46662, 1, 3879) - 1)/(V(46662, 1, 9) - 1) 18069 x25 Dec 2012 Lehmer primitive part
8(V(561, 1, 6309) + 1)/(V(561, 1, 9) + 1) 17319 x25 May 2016 Lehmer primitive part
9(V(21151, 1, 3777) - 1)/(V(21151, 1, 3) - 1) 16324 x25 May 2011 Lehmer primitive part
10(U(9275, 1, 3961) + U(9275, 1, 3960))/(U(9275, 1, 45) + U(9275, 1, 44)) 15537 x38 May 2009 Lehmer primitive part
11(V(824, 1, 5277) - 1)/(V(824, 1, 3) - 1) 15379 x25 Jan 2013 Lehmer primitive part
12(V(42995, 1, 3231) + 1)/(V(42995, 1, 9) + 1) 14929 x25 Nov 2012 Lehmer primitive part
13(V(8003, 1, 3771) + 1)/(V(8003, 1, 9) + 1) 14685 x25 Apr 2013 Lehmer primitive part
14(V(5111, 1, 3789) + 1)/(V(5111, 1, 9) + 1) 14019 x25 Feb 2013 Lehmer primitive part
15(V(5763, 1, 3753) + 1)/(V(5763, 1, 27) + 1) 14013 x25 May 2011 Lehmer primitive part
16(V(5132, 1, 3753) + 1)/(V(5132, 1, 27) + 1) 13825 x25 May 2011 Lehmer primitive part
17(V(4527, 1, 3771) + 1)/(V(4527, 1, 9) + 1) 13754 x25 Feb 2013 Lehmer primitive part
18(V(3813, 1, 3771) - 1)/(V(3813, 1, 9) - 1) 13473 x25 May 2011 Lehmer primitive part
19(V(3476, 1, 3771) - 1)/(V(3476, 1, 9) - 1) 13322 x25 May 2011 Lehmer primitive part
20(V(3755, 1, 3753) - 1)/(V(3755, 1, 27) - 1) 13319 x25 May 2011 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."]
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