PreprintCC BY-NC-SA 4.0arXiv
Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
Abstract
A set {a,b,c} of distinct positive integers is called a Diophantine triple if ab+1, ac+1, and bc+1 are perfect squares. Dujella formulated the conjecture that the only integral x-coordinates on the attached elliptic curve y2=(ax+1)(bx+1)(cx+1) are 0,d-,d+, together with −1 when the triple contains $1$, where d±=a+b+c+2abc ±2√((ab+1)(ac+1)(bc+1)). The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with x≠−1 make ax+1, bx+1, and cx+1 all perfect squares? We construct infinitely many triples {5,115,c} admitting an integral point with x≠−1 for which all three factors are nonsquares.
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BibTeX
@misc{jurasic2026counterexamples,
title = {{Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple}},
author = {Ana Jurasić and Matej Jurasić},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22635},
eprint = {2608.22635},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22635},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
AU - Ana Jurasić
AU - Matej Jurasić
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.22635
UR - https://arxiv.org/abs/2608.22635
AB - A set {a,b,c} of distinct positive integers is called a Diophantine triple if ab+1, ac+1, and bc+1 are perfect squares. Dujella formulated the conjecture that the only integral x-coordinates on the attached elliptic curve y^2=(ax+1)(bx+1)(cx+1) are 0,d_-,d_+, together with −1 when the triple contains $1$, where d_±=a+b+c+2abc ±2√((ab+1)(ac+1)(bc+1)). The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with x≠−1 make ax+1, bx+1, and cx+1 all perfect squares? We construct infinitely many triples {5,115,c} admitting an integral point with x≠−1 for which all three factors are nonsquares.
ER -