The Top Twenty--a Prime Page Collection

Lehmer number

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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. Comments and suggestions requested.

(up) 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
Un(P,Q) = (an - bn)/(a - b),     and     Vn(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:
definition of Lehmer numbers
These share many properties with the generalized Lucas numbers. 

For integers P, the two numbers Uk+1(P,1) + Uk(P,1) are Lehmer numbers whose product is equal to U2k+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 Uk+1(P,1) + Uk(P,1) cannot be prime if 2k+1 is composite.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1U(16531, 1, 6721)-U(16531, 1, 6720) 28347 x36 May 2007 Lehmer number
2U(15631, 1, 5040)-U(15631, 1, 5039) 21134 x25 Apr 2003 Lehmer number
3U(9657, 1, 4321)-U(9657, 1, 4320) 17215 x23 Dec 2005 Lehmer number
4U(15823, 1, 3960)-U(15823, 1, 3959) 16625 x25 Nov 2002 Lehmer number, cyclotomy
5U(10803, 1, 4081)-U(10803, 1, 4080) 16457 x25 Dec 2005 Lehmer number, cyclotomy
6U(10853, 1, 3960)+U(10853, 1, 3959) 15977 x25 Dec 2002 Lehmer number, cyclotomy
7U(9667, 1, 3960)-U(9667, 1, 3959) 15778 x25 Nov 2002 Lehmer number, cyclotomy
8U(8747, 1, 3780)+U(8747, 1, 3779) 14897 x25 Dec 2005 Lehmer number
9U(25700, 1, 3360)+U(25700, 1, 3359) 14813 x25 Jan 2004 Lehmer number, cyclotomy
10U(4951, 1, 3960)-U(4951, 1, 3959) 14628 CH3 Oct 2005 Lehmer number
11U(5192, 1, 3841)-U(5192, 1, 3840) 14267 x23 Nov 2005 Lehmer number
12U(3865, 1, 3960)+U(3865, 1, 3959) 14202 x25 Oct 2002 Lehmer number, cyclotomy
13U(3645, 1, 3841)-U(3645, 1, 3840) 13677 x25 Jul 2005 Lehmer number
14U(7644, 1, 3421)-U(7644, 1, 3420) 13281 CH3 Sep 2005 Lehmer number
15U(10206, 1, 3276)-U(10206, 1, 3275) 13130 x23 Dec 2005 Lehmer number
16U(12159, 1, 3150)-U(12159, 1, 3149) 12864 x25 Dec 2005 Lehmer number, cyclotomy
17U(5485, 1, 3421)+U(5485, 1, 3420) 12788 CH3 Sep 2005 Lehmer number
18U(6393, 1, 3276)-U(6393, 1, 3275) 12464 x25 Sep 2005 Lehmer number, cyclotomy
19U(4857, 1, 3300)-U(4857, 1, 3299) 12162 x25 Jul 2005 Lehmer number, cyclotomy
20U(5989, 1, 3169)-U(5989, 1, 3168) 11967 x25 Jul 2005 Lehmer number

(up) References

Gyory1982
Györy, K., "On some arithmetical properties of Lucas and Lehmer numbers," Acta Arith., 40:4 (1981/82) 369--373.  MR667047
Gyory2003
Györy, K., "On some arithmetical properties of Lucas and Lehmer numbers. II," Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.), 30 (2003) 67--73.  Dedicated to the memory of Professor Dr. P\'eter Kiss.  MR2054716
LP2003
Luca, F. and Porubský, S., "The multiplicative group generated by the Lehmer numbers," Fibonacci Quart., 41:2 (2003) 122--132.  MR1990520
McDaniel1993
McDaniel, W., "Square Lehmer numbers," Colloq. Math., 66:1 (1993) 85--93.  MR1242648
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.]
RW1980
Rotkiewicz, A. and Wasén, R., "Lehmer's numbers," Acta Arith., 36:3 (1980) 203--217.  MR581371
Schinzel1962
Schinzel, A., "On primitive prime factors of Lehmer numbers. I," Acta. Arith., 8 (1962/1963) 213--223.  MR 27:1408
Schinzel1962b
Schinzel, A., "The intrinsic divisors of Lehmer numbers in the case of negative discriminant," Ark. Mat., 4 (1962) 413--416 (1962).  MR0139567
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
SS1981
Shorey, T. N. and Stewart, C. L., "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. II," J. London Math. Soc. (2), 23:1 (1981) 17--23.  MR 82m:10025
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
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
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
Ward1955
Ward, M., "The intrinsic divisors of Lehmer numbers," Ann. of Math. (2), 62 (1955) 230--236.  MR0071446
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