2017/05/25 by Alan Filipin, Filipin, Alan, Ana Jurasić +1 · 1 citation
Mathematics · #11D09 #11D45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1705.09194
openalex publication_date 2017/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that every Diophantine quadruple in ℝ[X] is regular. More precisely, we prove that if \a, b, c, d\ is a set of four non-zero polynomials from ℝ[X], not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from ℝ[X], then (a+b-c-d)2=4(ab+1)(cd+1). One consequence of this result is that there does not exist a set of four non-zero polynomials from ℤ[X], not all constant, such that a product of any two of them increased by a positive integer n, which is not a perfect square, is a square of a polynomial from ℤ[X]. Our result also implies that there does not exist a set of five non-zero polynomials from ℤ[X], not all constant, such that a product of any two of them increased by a positive integer n, which is a perfect square, is a square of a polynomial from ℤ[X].