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Arithmetic properties of polynomials. (English) Zbl 1474.11088

Summary: First, we prove that the Diophantine system \[f (z) = f (x) + f (y) = f (u) - f (v) = f (p) f (q)\] has infinitely many integer solutions for \(f (X) = X(X +a)\) with nonzero integers \(a \equiv 0, 1, 4 \; (\bmod \; 5)\). Second, we show that the above Diophantine system has an integer parametric solution for \(f (X) = X(X + a)\) with nonzero integers \(a\), if there are integers \(m, n, k\) such that \[(n^2 - m^2)(4mnk(k + a + 1) + a(m^2 + 2mn - n^2)) \equiv 0 \; (\bmod \; (m^2 + n^2)^2),\] \[(m^2 + 2mn -n^2)((m^2 - 2mn - n^2)k(k + a + 1) - 2amn) \equiv 0 \; (\bmod \; (m^2 + n^2)^2),\] where \(k \equiv 0 \; (\bmod \; 4)\) when \(a\) is even, and \(k \equiv 2 \; (\bmod \; 4)\) when \(a\) is odd. Third, we get that the Diophantine system \[f (z) = f (x) + f (y) = f (u) - f (v) = f (p) f (q) = \frac{f (r)}{f (s)}\] has a five-parameter rational solution for \(f (X) = X(X + a)\) with nonzero rational number \(a\) and infinitely many nontrivial rational parametric solutions for \(f (X) = X(X +a)(X +b)\) with nonzero integers \(a, b\) and \(a \ne b\). Finally, we raise some related questions.

MSC:

11D25 Cubic and quartic Diophantine equations
11D72 Diophantine equations in many variables
11G05 Elliptic curves over global fields
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