Prime Numbers of the Form $24m+1$ and Erdös-Straus Conjecture

Source abstract

In 1948, Erdös and Straus formulated a conjecture on Egyptian fractions. One only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. In this article, we give a necessary and sufficient polynomial condition for a number of the form $24m+1$ to be a prime number. Then we present 56 solvability polynomials consisting of our earlier tame solutions, wild solutions and the polynomials for composite numbers of the form $24m+1$. Numeric test shows that they imply the conjecture. Moreover, we give a list of 36 exponential-polynomial functions consisting of our earlier 28 tame solutions and the polynomials for composite numbers, whose numeric test shows that Howerton's conjecture on fourteen wild primes holds. Furthermore, we find a version of Goldbach conjecture modulo 24.