Lehmer number

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This page is about one of those forms.

Definitions and Notes

Lucas generalized the sequence of Fibonacci numbers as follows: 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 set

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

Lehmer noted that we can loosen the restriction that P be an integer and still get a sequence of integers by replacing P with the squareroot of R and slightly modifying these definitions.  To make sure the sequences are not zero infinitely often we require that a/b not be a root of unity:

These share many properties with the generalized Lucas numbers. 

For integers P, the two numbers U_k+1(P,1) ± U_k(P,1) are Lehmer numbers whose product is equal to U_2k+1(P,1).  This follows from the fact that when Q=1 we may write (P ± sqrt(P²-4Q))/2 as the square of (sqrt(P+2) ± sqrt(P-2))/2, and hence obtain Lehmer's sqrt(R) as sqrt(P+2).

Note that the Lehmer numbers U_k+1(P,1) ± U_k(P,1) cannot be prime if 2k+1 is composite.

Record Primes of this Type

rank prime digits who when comment 1 U(15694, 1, 14700) + U(15694, 1, 14699) 61674 x45 Aug 2019 Lehmer number 2 U(809, 1, 17325) - U(809, 1, 17324) 50378 x45 Jul 2019 Lehmer number 3 U(52245, 1, 9241) + U(52245, 1, 9240) 43595 x45 Jul 2019 Lehmer number 4 U(35896, 1, 7260) + U(35896, 1, 7259) 33066 x45 Jul 2019 Lehmer number 5 U(23396, 1, 6615) + U(23396, 1, 6614) 28898 x45 Jul 2019 Lehmer number 6 U(16531, 1, 6721) - U(16531, 1, 6720) 28347 x36 May 2007 Lehmer number 7 U(5092, 1, 7561) + U(5092, 1, 7560) 28025 x25 Oct 2014 Lehmer number 8 U(5239, 1, 7350) - U(5239, 1, 7349) 27333 CH10 Jun 2017 Lehmer number 9 U(1766, 1, 7561) - U(1766, 1, 7560) 24548 x25 Nov 2013 Lehmer number 10 U(1383, 1, 7561) + U(1383, 1, 7560) 23745 x25 Nov 2013 Lehmer number 11 U(1118, 1, 7561) - U(1118, 1, 7560) 23047 x25 Oct 2013 Lehmer number 12 U(43100, 1, 4620) + U(43100, 1, 4619) 21407 x25 Jun 2016 Lehmer number 13 U(15631, 1, 5040) - U(15631, 1, 5039) 21134 x25 Apr 2003 Lehmer number 14 U(35759, 1, 4620) + U(35759, 1, 4619) 21033 x25 May 2016 Lehmer number 15 U(31321, 1, 4620) - U(31321, 1, 4619) 20767 x25 Jun 2016 Lehmer number 16 U(22098, 1, 4620) + U(22098, 1, 4619) 20067 x25 Jun 2016 Lehmer number 17 U(21412, 1, 4620) - U(21412, 1, 4619) 20004 x25 Jun 2016 Lehmer number 18 U(19361, 1, 4620) + U(19361, 1, 4619) 19802 x25 May 2016 Lehmer number 19 U(9657, 1, 4321) - U(9657, 1, 4320) 17215 x23 Dec 2005 Lehmer number 20 U(15823, 1, 3960) - U(15823, 1, 3959) 16625 x25 Nov 2002 Lehmer number, cyclotomy

References

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

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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

Ward1955

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