On the generalized Ramanujan-Nagell equation $$\pmb {x^2+(3m^2+1)=(4m^2+1)^n}$$ x 2 + ( 3 m 2 + 1 ) = ( 4 m 2 + 1 ) n

Ruiqin Fu () and Hai Yang ()
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Ruiqin Fu: Xi’an Shiyou University
Hai Yang: Xi’an Polytechnic University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 1, 222-227

Abstract: Abstract Let m be a positive integer. Using certain properties of Pell equations with elementary number theoretic methods, we prove that the equation $$x^2+(3m^2+1)=(4m^2+1)^n$$ x 2 + ( 3 m 2 + 1 ) = ( 4 m 2 + 1 ) n has only two positive integer solutions $$(x,\ n)=(m,\ 1)$$ ( x , n ) = ( m , 1 ) and $$(8m^3+3m,\ 3)$$ ( 8 m 3 + 3 m , 3 ) .

Keywords: exponential diophantine equation; generalized Ramanujan-Nagell equation; Pell equation; 11D61 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00209-2

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