The Diophantine equation px+6y=z2n: a complete solution for half of the primes
Abstract
Let p be a prime number and let n≥ 1 be an integer. We completely solve the exponential Diophantine equation px+6y=z2n in positive integers x,y,z for every prime p such that the Legendre symbol leg6p equals −1, that is, for p≡ 7,11,13,17pmod24: the equation has no solution, with the single exception (p,n,x,y,z)=(13,1,1,2,7). Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent n. Two ingredients of the proof have independent interest. First, we show that the Ramanujan--Nagell type equation px=2· 6m+1 has no solution with x≥ 2, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case n=1 and solves the open problems formulated there. Secondly, we prove that 13x+6y=z2 has (x,y,z)=(1,2,7) as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with leg6p=+1 realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when leg6p=+1.
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BibTeX
@misc{tiebekabe2026diophantine,
title = {{The Diophantine equation $p^x+6^y=z^2n$: a complete solution for half of the primes}},
author = {Pagdame Tiebekabe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09214},
eprint = {2609.09214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09214},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Diophantine equation p^x+6^y=z^2n: a complete solution for half of the primes
AU - Pagdame Tiebekabe
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.09214
UR - https://arxiv.org/abs/2609.09214
AB - Let p be a prime number and let n≥ 1 be an integer. We completely solve the exponential Diophantine equation p^x+6^y=z^2n in positive integers x,y,z for every prime p such that the Legendre symbol leg6p equals −1, that is, for p≡ 7,11,13,17pmod24: the equation has no solution, with the single exception (p,n,x,y,z)=(13,1,1,2,7). Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent n. Two ingredients of the proof have independent interest. First, we show that the Ramanujan--Nagell type equation p^x=2· 6^m+1 has no solution with x≥ 2, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case n=1 and solves the open problems formulated there. Secondly, we prove that 13^x+6^y=z^2 has (x,y,z)=(1,2,7) as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with leg6p=+1 realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when leg6p=+1.
ER -