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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 Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

We show that the system of equations t x + t y = t p , t y + t z = t q , t x + t z = t r , where t x = x ( x + 1 ) / 2 is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system t x + t y = t p , t y + t z = t q , t x + t z = t r , t x + t y + t z = t s has infinitely many rational two-parameter solutions.

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.

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