The One-Power Erdős Conjecture for Primes p ≡ 7 pmod36
Abstract
A conjecture of Erdős asserts that every odd integer n > 1 can be written as n = 2a + m, where a ≥ 0 is an integer and m is a squarefree positive integer. We study this conjecture for primes p ≡ 7 pmod36. 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 ≥ 5. For such q a separation principle applies --- q2 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 ≤ 3 have relative density 1 − ρ4, where ρ4 = 1.8637 × 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 108, always with a ≤ 5, and we exhibit the eleven primes below 108 requiring a = 5. An appendix corrects the error bound in a lemma of Hercher, whose conclusions are unaffected.
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BibTeX
@misc{harman2026onepower,
title = {{The One-Power Erdős Conjecture for Primes $p \equiv 7 \pmod36$}},
author = {Shen Harman and Dana Paquin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07214},
eprint = {2610.07214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07214},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The One-Power Erdős Conjecture for Primes p ≡ 7 pmod36
AU - Shen Harman
AU - Dana Paquin
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07214
UR - https://arxiv.org/abs/2610.07214
AB - A conjecture of Erdős asserts that every odd integer n > 1 can be written as n = 2^a + m, where a ≥ 0 is an integer and m is a squarefree positive integer. We study this conjecture for primes p ≡ 7 pmod36. 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 ≥ 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 ≤ 3 have relative density 1 − ρ_4, where ρ_4 = 1.8637 × 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 ≤ 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.
ER -