A344202 Primes p such that gcd(ord_p(2), ord_p(3)) = 1.

683, 599479, 108390409, 149817457, 666591179, 2000634731, 4562284561, 14764460089, 24040333283, 2506025630791, 5988931115977

Offset: 1

Author

Sofia Lacerda, May 11 2021

Keywords

• nonn
• more
• hard

Comments

'ord_p' here means the multiplicative order, not to be confused with the p-adic order that is also often denoted by ord_p.

Related to Diophantine equations of the form (2^x-1)*(3^y-1) = n*z^2.

Links

• Sofia Lacerda, C++ program (github).
• MathOverflow, Solve this Diophantine equation (2^x-1)(3^y-1)=2z^2
• Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021.
• Wikipedia, Multiplicative order.
• Wikipedia, P-adic_valuation.

Crossrefs

Cf. A014664, A062117.

Programs

Mathematica

Select[Range[10^6], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[3, #]] &] (* Amiram Eldar, May 11 2021 *)

PARI

isok(p) = isprime(p) && (gcd(znorder(Mod(2, p)), znorder(Mod(3, p))) == 1); \\ Michel Marcus, May 11 2021

Python

from sympy.ntheory import n_order
from sympy import gcd, nextprime
A344202_list, p = [], 5
while p < 10**9:
    if gcd(n_order(2,p),n_order(3,p)) == 1:
        A344202_list.append(p)
    p = nextprime(p) # Chai Wah Wu, May 12 2021

Extensions

a(3)-a(5) from Michel Marcus, May 11 2021

a(6)-a(8) from Amiram Eldar, May 11 2021

a(9) from Daniel Suteu, May 16 2021

a(10) from Sofia Lacerda, Jul 07 2021

a(11) from Sofia Lacerda, Aug 03 2021