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A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ( a n ) x + ( b n ) y = ( c n ) z in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ( u 2 - v 2 ) x + ( 2 u v ) y = ( u 2 + v 2 ) z , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

A note on ternary purely exponential diophantine equations

Yongzhong Hu, Maohua Le (2015)

Acta Arithmetica

Let a,b,c be fixed coprime positive integers with mina,b,c > 1, and let m = maxa,b,c. Using the Gel’fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation a x + b y = c z satisfy maxx,y,z < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with minx,y,z > 1.

A note on the diophantine equation k 2 - 1 = q n + 1

Maohua Le (1998)

Colloquium Mathematicae

In this note we prove that the equation k 2 - 1 = q n + 1 , q 2 , n 3 , has only finitely many positive integer solutions ( k , q , n ) . Moreover, all solutions ( k , q , n ) satisfy k 10 10 182 , q 10 10 165 and n 2 · 10 17 .

A note on the diophantine equation x 2 + b Y = c z

Maohua Le (2006)

Czechoslovak Mathematical Journal

Let a , b , c , r be positive integers such that a 2 + b 2 = c r , min ( a , b , c , r ) > 1 , gcd ( a , b ) = 1 , a is even and r is odd. In this paper we prove that if b 3 ( m o d 4 ) and either b or c is an odd prime power, then the equation x 2 + b y = c z has only the positive integer solution ( x , y , z ) = ( a , 2 , r ) with min ( y , z ) > 1 .

A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 - D = p n

Yuan-e Zhao, Tingting Wang (2012)

Czechoslovak Mathematical Journal

Let D be a positive integer, and let p be an odd prime with p D . In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N ( D , p ) , and also prove that if the equation U 2 - D V 2 = - 1 has integer solutions ( U , V ) , the least solution ( u 1 , v 1 ) of the equation u 2 - p v 2 = 1 satisfies p v 1 , and D > C ( p ) , where C ( p ) is an effectively computable constant...

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