cp's OEIS Frontend

A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).

%I A218028 #44 Oct 12 2024 02:52:40
%S A218028 2,3,10,12,33,18,10,9,12,8,4,60,5,85,70,45,31,79,92,170,43,76,152,59,
%T A218028 59,139,256,64,62,40,44,188,177,18,14,156,227,192,231,223,79,31,75,
%U A218028 362,7,239,338,402,6,235,114,72,342,511,15,483,310,355,104,292,232
%N A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).
%C A218028 A007519(n) : primes of form 8n+1.
%H A218028 Robert Israel, <a href="/A218028/b218028.txt">Table of n, a(n) for n = 1..10000</a>
%H A218028 Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, <a href="https://doi.org/10.7546/nntdm.2024.30.3.516-529">The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences</a>, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
%e A218028 a(5) = 33 because 33^4+1 = 1185922 = 2 * 97 * 6113 with A007519(5) = 97.
%p A218028 V:= Vector(100): count:= 0:
%p A218028 for p from 9 by 8 while count < 100 do
%p A218028   if isprime(p) then
%p A218028       count:= count+1; V[count]:=min(map(rhs@op,[msolve(k^4+1,p)]))
%p A218028     fi
%p A218028 od:
%p A218028 convert(V,list); # _Robert Israel_, Mar 13 2018
%t A218028 aa = {}; Do[p = Prime[n]; If[Mod[p, 8] == 1, k = 1; While[ ! Mod[k^4 + 1, p] == 0, k++ ]; AppendTo[aa, k]], {n, 300}]; aa
%Y A218028 Cf. A017077, A007519.
%K A218028 nonn
%O A218028 1,1
%A A218028 _Michel Lagneau_, Oct 22 2012