Some meromorphic functions associated to the S-unit equation
Some observations on the Diophantine equation f(x)f(y) = f(z)²
Let f ∈ ℚ [X] be a polynomial without multiple roots and with deg(f) ≥ 2. We give conditions for f(X) = AX² + BX + C such that the Diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial integer solutions and prove that this equation has a rational parametric solution for infinitely many irreducible cubic polynomials. Moreover, we consider f(x)f(y) = f(z)² for quartic polynomials.
Some problems in multidimensional analytic number theory
Some problems involving powers of integers
Some problems of Waring?s type
Some Recent Developments in three Classical Problems of Number Theory.
Some remarks on Fermat's conjecture
Some remarks on the equation in Fibonacci numbers.
Some remarks on the S-unit equation in function fields
Some Remarks on Vinogradov's Mean Value Theorem and Tarry's Problem.
Some results and problems in number theory
Some results concerning the nearest integer continued fraction expansion of ...D.
Some reults on fundamental units in cubic fields.
Some simple elliptic surfaces of genus zero.
Some special curves of genus 5
Some variations and consequences of the Kummer-Mirimanoff congruences
Sommes de carrés dans [Book]
Sophie Germain's principle and Lucas numbers.
Součty celých v lomené arithmetické posloupnosti. [I.]
Sparsity of the intersection of polynomial images of an interval
We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies , where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then .