A note on a conjecture of Jeśmanowicz

Tom 86 / 2000

Moujie Deng, G. Cohen Colloquium Mathematicum 86 (2000), 25-30 DOI: 10.4064/cm-86-1-25-30


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 $(an)^x+(bn)^y=(cn)^z$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation $(u^2-v^2)^x+(2uv)^y=(u^2+v^2)^z$, 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.


  • Moujie Deng
  • G. Cohen

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