2,431 research outputs found
The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z
This paper answers a question asked by Ed Pegg Jr. in 2001: "What is the
maximal value of min(A,B,C)/ gcd(A,B,C) for A^x + B^y = C^z with A,B,C >= 1;
x,y,z >= 3?" Equations of this form are analyzed, showing how they map to
exponential Diophantine equations with coprime bases. A search algorithm is
provided to find the largest min/gcd value within a given equation range. The
algorithm precalculates a multi-gigabyte lookup table of power residue
information that is used to eliminate over 99% of inputs with a single array
lookup and without any further calculations. On inputs that pass this test, the
algorithm then performs further power residue tests, avoiding modular powering
by using lookups into precalculated tables, and avoiding division by using
multiplicative inverses. This algorithm is used to show the largest min/gcd
value for all equations with C^z <= 2^100.Comment: Body: 16 pages, Appendices: 11 pages, 5 tables, 1 figur
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
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