login
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).
1

%I #44 Oct 12 2024 02:52:40

%S 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 59,139,256,64,62,40,44,188,177,18,14,156,227,192,231,223,79,31,75,

%U 362,7,239,338,402,6,235,114,72,342,511,15,483,310,355,104,292,232

%N 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 A007519(n) : primes of form 8n+1.

%H Robert Israel, <a href="/A218028/b218028.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(5) = 33 because 33^4+1 = 1185922 = 2 * 97 * 6113 with A007519(5) = 97.

%p V:= Vector(100): count:= 0:

%p for p from 9 by 8 while count < 100 do

%p if isprime(p) then

%p count:= count+1; V[count]:=min(map(rhs@op,[msolve(k^4+1,p)]))

%p fi

%p od:

%p convert(V,list); # _Robert Israel_, Mar 13 2018

%t 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 Cf. A017077, A007519.

%K nonn

%O 1,1

%A _Michel Lagneau_, Oct 22 2012