NECESSARY CONDITIONS FOR EXISTENCE OF A FRIEND OF 3

Abstract

This paper sets out to, systematically, use properties of the abundancy index function to prove that a friend ݉ of 38 must be an odd non-square multiple of 19ଶwhich is not divisible by 3, and that every prime factor ݍ of ݉ such that 4|(ݍ + 1)has an even exponent in the prime factorization of  ݉.  In addition, if the power of 19 in  ݉  is 2, then  127|݉  in which case the power of 127 must be even, larger than 2 and not equal to 8, and if the power of 19 is 6, both 701 and 70841 would be compulsory prime factors of  ݉, where the power of 701 cannot equal 1 or 3. The paper also establishes that it is not possible to have 8 as the power of 19 in the prime factorization of ݉