Siar–Keskin bounded-exponent conjecture for (an2)(bn2)=x2(a^n-2)(b^n-2)=x^2

Let a,b,n,xa,b,n,x be natural numbers with

2<a<b.2<a<b.

Consider the exponential Diophantine equation

(an2)(bn2)=x2.(a^n-2)(b^n-2)=x^2.

Siar–Keskin's bounded-exponent conjecture. If this equation has a solution n,xn,x, then

n6.n\leq 6.

The conjecture concerns a uniform bound on the exponent in this family of exponential Diophantine equations. The source gives no proof or resolution, so the claim remains open.

Sources & referencesView supporting material

Primary source

Zafer Şiar and Refik Keskin, “On the Exponential Diophantine Equation (a^2-2)(b^2-2)=x^2”, arXiv:1801.04770 (2018).

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