Elliptic binomial diophantine equations

A Binomial Diophantine Equation

following result. THEOREM 1. The only (n,m)eZ with n^2 and m5=4 satisfying © = ( 7 ) a r e {n> m)=(2> 4)> (6> 6)> and (21> Our binomial diophantine equation represents an elliptic curve, since it can be rewritten as a quartic polynomial being a square. Indeed, on putting u = 2/i 1 and v = 2m 3, we see at once that Theorem 1 follows from the following result. THEOREM 2. The only (u, v) e Z with ...

On a Few Diophantine Equations, in Particular, Fermat’s Last Theorem

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat’s last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equation...

On the number of solutions of simultaneous Pell equations

It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x − az = 1, y − bz = 1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solutions, this result is not too far from the truth. If, further, u and v are nonzero integers with av − b...

Combinatorial Diophantine Equations and a Refinement of a Theorem on Separated Variables Equations

We look at Diophantine equations arising from equating classical counting functions such as perfect powers, binomial coefficients and Stirling numbers of the first and second kind. The proofs of the finiteness statements that we give use a variety of methods from modern number theory, such as effective and ineffective tools from Diophantine approximation. As a tool for one part of the statement...

On the Elliptic Logarithm Method for Elliptic Diophantine Equations: Reflections and an Improvement