On the equation
On the exponential diophantine equation
For any positive integer which is not a square, let be the least positive integer solution of the Pell equation and let denote the class number of binary quadratic primitive forms of discriminant . If satisfies and , then is called a singular number. In this paper, we prove that if is a positive integer solution of the equation with , then maximum and both , are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions .
On the exponential Diophantine equations of the form .
On the exponential local-global principle
Skolem conjectured that the "power sum" A(n) = λ₁α₁ⁿ + ⋯ + λₘαₘⁿ satisfies a certain local-global principle. We prove this conjecture in the case when the multiplicative group generated by α₁,...,αₘ is of rank 1.
On the extendibility of the Diophantine triple .
On the extension of the Diophantine pair in .
On the family of diophantine triples .
On the family of Thue equations x³ - (n-1)x²y - (n+2)xy² - y³ = k
On the first case of Fermat's last theorem.
On the First Case of Fermat's Last Theorem, II.
On the Frobenius number of a modular Diophantine inequality
We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality , with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.
On the Frobenius number of a proportionally modular Diophantine inequality.
On the Frobenius number of Fibonacci numerical semigroups.
On the Frobenius number of some Lucas numerical semigroups.
On the Frobenius problem for .
On the Frobenius problem for geometric sequences.
On the Galois groups of cubics and trinomials
On the generalized Fermat equation over totally real fields
In a recent paper, Freitas and Siksek proved an asymptotic version of Fermat’s Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form , where A, B, C are odd integers belonging to a totally real field.
On the generalized Ramanujan-Nagell equation I
On the generalized Ramanujan-Nagell equation, II