On a class of Diophantine equations.
On a few Diophantine equations, in particular, Fermat's Last Theorem.
On certain combinatorial Diophantine equations and their connection to Pythagorean numbers
On so that .
On sot hat represents and and not .
On Diophantine quintuples
On Eisenstein's problem
On fundamental solutions of binary quadratic form equations
We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation Au²+Buv+Cv²=N, where A>0, N≠0 and B²-4AC is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation u²-dv²=N, where d is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.
On integer solutions to x² - dy² = 1, z² - 2dy² =1
On Obláth's problem.
On pythagorean angles
On Rédei's theory of the Pell equation.
On Shanks' algorithm for computing the continued fraction of .
On some properties of two simultaneous polygonal sequences.
On some special forms of simultaneous Pell equations
On Tate-Shafarevich groups of .
On ternary quadratic forms over the rational numbers
For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.
On the congruence f(x) + g(y) + c ≡ 0 (mod xy) (completion of Mordell's proof)
Assertions on the congruence f(x) + g(y) + c ≡ 0 (mod xy) made without proof by Mordell in his paper in Acta Math. 88 (1952) are either proved or disproved.
On the Diophantine equation
In this study, we determine when the Diophantine equation has an infinite number of positive integer solutions and for Moreover, we give all positive integer solutions of the same equation for in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation .
On the diophantine equation