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A generalization of a result of Fermat. (English) Zbl 1164.11002
It is an immediate consequence of the quadratic reciprocity law that no positive divisor of $$3^n+1$$ has the form $$12k+11$$. This result is generalized as follows: let $$p \equiv 1 \bmod 4$$ be prime; then no positive divisor of $$p^n+1$$ has the form $$4pk+a$$ if and only if one of the following conditions holds: i) $$p \mid a$$; ii) $$4 \mid a$$; iii) $$a \equiv 3 \bmod 4$$ and $$(\frac ap) = 1$$. The proof uses the quadratic reciprocity law for one direction, and Dirichlet’s theorem on primes in arithmetic progressions for the other one.
##### MSC:
 11A15 Power residues, reciprocity 11N13 Primes in congruence classes
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