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A343768
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Primes p such that p^2 + 1 divides 3^(p+1) - 1.
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0
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OFFSET
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1,1
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COMMENTS
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a(n) != 13 (mod 20) and a(n) != 17 (mod 20). Proof: Write n=20*k + 13, and then show that n^2 + 1 == 0 (mod 5) and 3^(n+1) - 1 == 3 (mod 5). Because of its factor 5 the former cannot divide the latter. - Martin Ehrenstein, Jun 06 2021
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LINKS
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MATHEMATICA
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a[n_Integer] := Select[ Prime@ Range@n, PowerMod[3, # + 1, #^2 + 1] == 1 &]; a[2*10^5] (* or *)
a[n_Integer] := If[ PrimeQ@n && Divisible[3^(n + 1) - 1, n^2 + 1] , Print@n]; Do[ a[n], {n, 2*10^5}]
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PROG
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(Python)
from sympy import primerange
def afind(limit):
for p in primerange(1, limit+1):
if pow(3, p+1, p**2+1) == 1: print(p, end=", ")
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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