A196627 - OEIS

%I #21 Jul 17 2019 09:09:54

%S 5,5333,18414001,804146449,1117131202441,11170373961701,

%T 92009554676141,110451862680769

%N Primes p such that the system of congruences { 2^x == 3 (mod p), 3^x == 2 (mod p) } has a solution.

%C Prime divisors of the elements of A109768.

%C The corresponding smallest positive solutions x are given by A196628.

%C Some larger terms: 600827908214213, 18969653181299397175271, 1098445767808750903973251, 364947672292511454405089069, 706132008101135602203621405289, 203315521506434771079581843014801, 29579867253585988507046633033646287, 183139575629088302014027581573180839, 59813046375181769306016700165290169537, 1517811599380242183731391003255018381040066573726286733611752067380771.

%C This sequence has density zero among all primes. More exactly, M. Skałba showed that the number of terms in this sequence below x is O(x/(log(x))^1.0243). -_Tomohiro Yamada_, Jul 17 2019

%H A. S. Izotov, <a href="http://www.fq.math.ca/Papers1/43-2/paper43-2-6.pdf">On prime divisors of GCD(3^n-2,2^n-3)</a>, Fibonacci Quarterly 43, May 2005, pp. 130-131.

%H M. Skałba, <a href="https://doi.org/10.1007/s00013-005-1115-6">Primes dividing both 2^n-3 and 3^n-2 are rare</a>, Archiv der Mathematik 84 (2005), issue 6, pp. 485-495.

%K nonn,hard,more

%O 1,1

%A _Max Alekseyev_, Oct 04 2011