The One-Power Erdős Conjecture for Primes $p\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}36)$

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 $p\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}36)$. Writing $x=(p-1)\mathrm{/}6$, 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 $1-{\rho }_4$, where ${\rho }_4=1.8637\times {10}^{-5}$ 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.