A062647 Numbers k such that 81^k - 80^k is prime.

3, 43, 113, 157, 269, 709, 1109, 2027, 8297, 86837, 310721

Offset: 1

Author

Mike Oakes, May 18 2001, May 19 2001

Keywords

• nonn
• hard
• more

Comments

Terms greater than 1000 are often only strong pseudoprimes.

a(11) > 10^5. - Robert Price, Jan 13 2015

Factors of these numbers are of the form p = 2*n*k + 1, just like the repunit numbers, but the PRP tests are significantly slower since there is not a fast mod() property like the repunits, where (number mod Rn) can be calculated with a folding of the digits at length n in base b. However, numbers of this form b^p - (b-1)^p seem to be prime with greater relative frequency than the repunits. While the repunits have a linear fit coefficient that approaches 0.56145948 (see link below for a Generalized Repunit Conjecture), this sequence currently has a linear fit coefficient of 0.22559. - Paul Bourdelais, Jul 06 2023

Links

• Paul Bourdelais, A Generalized Repunit Conjecture

Crossrefs

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

Programs

PARI

is(n)=ispseudoprime(81^n-80^n) \\ Charles R Greathouse IV, Jun 12 2017

Extensions

a(10) from Robert Price, Jan 13 2015 (Computer run by Adam Marciniec)

a(11) from Paul Bourdelais, Jul 06 2023