The One-Power Erdős Conjecture for Primes $p \equiv 7 \pmod{36}$

Abstract

A conjecture of Erdős asserts that every odd integer $n > 1$ can be written as $n = 2^a + m$, where $a \ge 0$ is an integer and $m$ is a squarefree positive integer. We study this conjecture for primes $p \equiv 7 \pmod{36}$. Writing $x = (p-1)/6$, the candidate remainders take the shapes $p - 1 = 6x$, $p-2 = 6x-1$, $p-4 = 3(2x-1)$, and $p-8 = 6x-7$, in which the primes $2$ and $3$ are completely controlled: every obstruction to squarefreeness comes from a prime $q \ge 5$. For such $q$ a separation principle applies --- $q^2$ divides at most one of the remainders, since two of them differ by at most $7 < 25$ --- and it persists across any window of exponents of width less than $20$. Separation collapses the square-divisor problem for the quartic $x(6x-1)(2x-1)(6x-7)$ into independent linear problems, and we deduce unconditionally that the primes of the class representable with $a \le 3$ have relative density $1 - ρ_4$, where $ρ_4 = 1.8637 \times 10^{-5}$ is an explicit Euler-type constant; in particular, infinitely many primes of the class are not so representable. Computationally, we verify the conjecture for all primes of the class below $10^8$, always with $a \le 5$, and we exhibit the eleven primes below $10^8$ requiring $a = 5$. An appendix corrects the error bound in a lemma of Hercher, whose conclusions are unaffected.