1,003 research outputs found
A new upper bound for odd perfect numbers of a special form
We shall given a new effectively computable upper bound of odd perfect
numbers whose Euler factors are powers of fixed exponent, improving our old
result in T. Yamada, Colloq. Math. 103 (2005), 303--307.Comment: 10 pages, the author's revised version; 3 pages, corrigendum to the
previous (published in the journal) versio
Searching for Diophantine quintuples
We consider Diophantine quintuples . These are sets of
distinct positive integers, the product of any two elements of which is one
less than a perfect square. It is conjectured that there are no Diophantine
quintuples; we improve on current estimates to show that there are at most
Diophantine quintuples.Comment: 15 page
On a conjecture on exponential Diophantine equations
We study the solutions of a Diophantine equation of the form ,
where , and . The main
result is that if there exists a solution with odd then
this is the only solution in integers greater than 1, with the possible
exception of finitely many values . We also prove the uniqueness of such
a solution if any of , , is a prime power. In a different vein, we
obtain various inequalities that must be satisfied by the components of a
putative second solution
A Survey on the Ternary Purely Exponential Diophantine Equation
Let , , be fixed coprime positive integers with .
In this survey, we consider some unsolved problems and related works concerning
the positive integer solutions of the ternary purely exponential
diophantine equation
On the Diophantine equation
In this paper we consider the Diophantine equation where
are integer unknowns with and are odd primes and
We prove that there are only finitely many solutions
for which is not a sum of two consecutive squares. We also
study the above equation with fixed and with fixed $q.
- β¦