A383041 Primes p such that for some y > 0, 2^y == 3^y == 2 (mod p).

331, 1210483, 45661129, 107889071, 13220372117, 452802040997, 2381120538437, 5836609770097, 6196278645943, 8322368214791

Offset

1,1

Comments

Equivalent to primes p where p | 3^n-2^n and p | 2^m-3^n, for some coprime m and n.

Motivated by analysis of the rational Collatz circuit of length m, with n odd terms, whose smallest member is (3^n-2^n)/(2^m-3^n) [Davison 1976].

For p = 13220372117, y = 7850019689; for 452802040997, y = 172800778897; for p = 2381120538437, y = 969305705913; for p = 5836609770097, y = 1574816965408; for p = 6196278645943, y = 783190158058; for p = 8322368214791, y = 12363639911.

References

J. L. Davison, Some Comments on an Iteration Problem, Proc. 6th Manitoba Conference on Numerical Mathematics and Computing (1976), 155-159.

R. P. Steiner, A Theorem on the Syracuse Problem, Proc. 7th Manitoba Conference on Numerical Mathematics and Computing (1978), 553-559.

Links

Table of n, a(n) for n=1..10.

M. Chamberland, An Update on the 3x+1 Problem, Butlleti de la Societat Catalana de Matematiques 18 (2003), 19-45. See p. 16 for (3^n-2^n)/(2^m-3^n) motivation.

Example

For p = 331, 2^121 = 3^121 = 2 (mod 331).

Mathematica

Select[Prime[Range[100]], With[{p=#}, AnyTrue[Range[p-1], Mod[{2, 3}^#, p] == {2, 2} &]] &]

Crossrefs

Sequence in context: A199820 A255389 A133141 * A231755 A097401 A376734

Adjacent sequences: A383038 A383039 A383040 * A383042 A383043 A383044

Keyword

nonn,more

Author

Kevin Knight, Apr 13 2025

Extensions

a(5)-a(10) added by Kevin Knight, May 15 2025

Status

approved