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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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