Geodesic
A remark on a Diophantine equation of S. S. Pillai
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 897-903
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S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x= a$, $\min (x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $\gcd (x,y)=1$.
S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x= a$, $\min (x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $\gcd (x,y)=1$.
DOI : 10.21136/CMJ.2024.0124-24
Classification : 11D61, 11D72
Keywords: Pillai's Diophantine equation; Lehmer sequence; primitive divisor 
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Hoque, Azizul. A remark on a Diophantine equation of S. S. Pillai. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 897-903. doi: 10.21136/CMJ.2024.0124-24

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