On the Exponent Average of Integer Sequences.
On the exponent of the ideal class groups of imaginary extensions of
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 Divisor Function
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 extendability of elliptic surfaces of rank two and higher
We study threefolds having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings...
On the extendibility of the Diophantine triple .
On the extension of the Diophantine pair in .
On the ...-factor attached to motives.
On the Factorization of p-adic L-series.
On the Factorization of the Relative Class Number in Terms of Frobenius Divisions.
On the factors of the period polynomial for finite fields
On the family of diophantine triples .
On the family of Thue equations x³ - (n-1)x²y - (n+2)xy² - y³ = k
On the Farey fractions with denominators in arithmetic progression.
On the Fermat periods of natural numbers.
On the Fibonacci -matrices of order .
On the finiteness of Pythagoras numbers of real meromorphic functions
We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...
On the finiteness of for motives associated to modular forms.