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

%I A344202 #53 May 03 2025 04:28:04
%S A344202 683,599479,108390409,149817457,666591179,2000634731,4562284561,
%T A344202 14764460089,24040333283,2506025630791,5988931115977
%N A344202 Primes p such that gcd(ord_p(2), ord_p(3)) = 1.
%C A344202 'ord_p' here means the multiplicative order, not to be confused with the p-adic order that is also often denoted by ord_p.
%C A344202 Related to Diophantine equations of the form (2^x-1)*(3^y-1) = n*z^2.
%H A344202 Sofia Lacerda, <a href="https://github.com/SofiaSL/ord23">C++ program</a> (github).
%H A344202 MathOverflow, <a href="https://mathoverflow.net/questions/296729/solve-this-diophantine-equation-2x-13y-1-2z2">Solve this Diophantine equation (2^x-1)(3^y-1)=2z^2</a>
%H A344202 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4190108/for-all-alpha-beta-in-mathbbn-are-there-only-finitely-many-primes-so-that">For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?</a>, 2021.
%H A344202 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_order">Multiplicative order</a>.
%H A344202 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_valuation">P-adic_valuation</a>.
%t A344202 Select[Range[10^6], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[3, #]] &] (* _Amiram Eldar_, May 11 2021 *)
%o A344202 (C++) // See Lacerda link.
%o A344202 (PARI) isok(p) = isprime(p) && (gcd(znorder(Mod(2, p)), znorder(Mod(3, p))) == 1); \\ _Michel Marcus_, May 11 2021
%o A344202 (Python)
%o A344202 from sympy.ntheory import n_order
%o A344202 from sympy import gcd, nextprime
%o A344202 A344202_list, p = [], 5
%o A344202 while p < 10**9:
%o A344202     if gcd(n_order(2,p),n_order(3,p)) == 1:
%o A344202         A344202_list.append(p)
%o A344202     p = nextprime(p) # _Chai Wah Wu_, May 12 2021
%Y A344202 Cf. A014664, A062117.
%K A344202 nonn,more,hard
%O A344202 1,1
%A A344202 _Sofia Lacerda_, May 11 2021
%E A344202 a(3)-a(5) from _Michel Marcus_, May 11 2021
%E A344202 a(6)-a(8) from _Amiram Eldar_, May 11 2021
%E A344202 a(9) from _Daniel Suteu_, May 16 2021
%E A344202 a(10) from _Sofia Lacerda_, Jul 07 2021
%E A344202 a(11) from _Sofia Lacerda_, Aug 03 2021