We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-12,
author = {J. Browkin and A. Schinzel},
title = {Prime factors of values of polynomials},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {135-138},
zbl = {1239.11102},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-12}
}
J. Browkin; A. Schinzel. Prime factors of values of polynomials. Colloquium Mathematicae, Tome 122 (2011) pp. 135-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-12/