%I #23 Feb 24 2023 19:32:27
%S 31,601,2593,20478961,204700049,668731841
%N Primes p that divide both 3^k-2 and 5^k-1 for some k.
%C If prime p divides 3^k-2 and 5^k-1, then p divides 3^j-2 and 5^j-1 for all j such that j == k (mod p-1).
%C Primes p such that the equation 3^(x*A070677(p)) == 2 (mod p) has a solution.
%C Values of k: 24, 108, 64, 376020, 67141466, 487515840, ... - _Chai Wah Wu_, Feb 24 2023
%e a(3) = 2593 is a term because 2593 is prime, 3^64 == 2 (mod 2593) and 5^64 == 1 (mod 2593).
%p R:= NULL: count:= 0: p:= 5: with(numtheory):
%p while count < 4 do
%p p:= nextprime(p);
%p if mlog(2,3 &^ order(5,p) mod p, p) <> FAIL then R:= R,p; count:= count+1 fi
%p od:
%p R;
%o (Python)
%o from itertools import islice
%o from sympy import discrete_log, nextprime, n_order
%o def A360761_gen(): # generator of terms
%o p = 5
%o while True:
%o try:
%o discrete_log(p:=nextprime(p),2,pow(3,n_order(5,p),p))
%o except:
%o continue
%o yield p
%o A360761_list = list(islice(A360761_gen(),4)) # _Chai Wah Wu_, Feb 23 2023
%Y Cf. A070677.
%K nonn,more
%O 1,1
%A _Robert Israel_, Feb 19 2023
%E a(5)-a(6) from _Chai Wah Wu_, Feb 23 2023