PreprintCC BY-NC-ND 4.0arXiv
On the Diophantine equation Cx2+p12m1p22m2⋯ pk2mk=yq
Abstract
Let C>3 be an odd square-free integer, q an odd prime not dividing the class number of ℚ(√(−C)), and p1,…,pk distinct odd primes not dividing C. We describe all solutions of Cx2+p12m1⋯ pk2mk=yq in positive integers with gcd(x,y)=1 and y odd. Writing N=p1m1⋯ pkmk, every solution comes from a unique factorization x√C+N√(−1)=±(u√C± v√(−1))q, or from a half-integral analogue when q=3, in which each pi≠ q divides exactly one of v and N/v. Using primitive divisors of Lehmer sequences, we show that for q≥5 some pi satisfies pi≡leg−Cpipmod q; in particular, there is no solution when q≥max(5,p1+2,…,pk+2). For q=3 the equation reduces to four explicit auxiliary equations.
Related
BibTeX
@misc{sangale2026diophantine,
title = {{On the Diophantine equation $Cx^2+p_1^2m_1p_2^2m_2\cdots p_k^2m_k=y^q$}},
author = {Usha K. Sangale and Priyanka R. Satpute},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04923},
eprint = {2611.04923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04923},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Diophantine equation Cx^2+p_1^2m_1p_2^2m_2⋯ p_k^2m_k=y^q
AU - Usha K. Sangale
AU - Priyanka R. Satpute
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04923
UR - https://arxiv.org/abs/2611.04923
AB - Let C>3 be an odd square-free integer, q an odd prime not dividing the class number of ℚ(√(−C)), and p_1,…,p_k distinct odd primes not dividing C. We describe all solutions of Cx^2+p_1^2m_1⋯ p_k^2m_k=y^q in positive integers with gcd(x,y)=1 and y odd. Writing N=p_1^m_1⋯ p_k^m_k, every solution comes from a unique factorization x√C+N√(−1)=±(u√C± v√(−1))^q, or from a half-integral analogue when q=3, in which each p_i≠ q divides exactly one of v and N/v. Using primitive divisors of Lehmer sequences, we show that for q≥5 some p_i satisfies p_i≡leg−Cp_ipmod q; in particular, there is no solution when q≥max(5,p_1+2,…,p_k+2). For q=3 the equation reduces to four explicit auxiliary equations.
ER -