A note on solitary numbers

Abstract

Does $14$ have a friend? Until now, this has been an open question. In this note, we prove that a potential friend $F$ of $14$ is an odd, non-square positive integer. $7$ appears in the prime factorization of $F$ with an even exponent while at most two prime divisors of $F$ can have odd exponents in the prime factorization of $F$. If $p\mid F$ such that $p$ is congruent to $7$ modulo $8$, then $p^{2a}\mid \mid F$, for some positive integer $a$. Further, no prime divisor of $F$ has an exponent congruent to $7$ modulo $8$ and no prime divisor can exceed $1.4\sqrt{F}$. The primes $3,5$ cannot appear simultaneously in the prime factorization of $F$. If $(3,F)>1$ or $(5,F)>1$, then $\ddot{I}‰(F)\ge 4$, otherwise $\ddot{I}‰(F)\ge 8$.

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