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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.
11A15 Power residues, reciprocity
11N13 Primes in congruence classes
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