A note on solitary numbers

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 | F such that p is congruent to 7 modulo 8, then p²ᵃ || 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√F. The primes 3, 5 cannot appear simultaneously in the prime factorization of F. If (3,F) > 1 or (5,F) > 1, then ω(F) ≥ 4, otherwise ω(F) ≥ 8.