The search session has expired. Please query the service again.
The search session has expired. Please query the service again.
The search session has expired. Please query the service again.
The search session has expired. Please query the service again.
In this paper, we study equations of the form , where is a binary form, homogeneous of degree , which is supposed to be primitive and irreducible, and is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...
We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system
has positive integer or rational solutions , , , , Using the theory of elliptic curves, we prove that it has no positive integer solution for , but there are infinitely many positive integers such that it has a positive integer solution for . As a corollary, for and any positive integer , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...
The famous problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations , with and , for all primes and indeed for all but primes . Here the strategy of [10] is not sufficient due to the sizes of...
Currently displaying 1 –
20 of
32