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On Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. (English) Zbl 1309.11030

Summary: In [Ann. Soc. Math. Pol., Ser. II, Wiad. Mat. 1, No. 2, 196–202 (1956; Zbl 0074.27205)] L. Jeśmanowicz conjectured that the exponential Diophantine equation \((m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\), where \(m\) and \(n\) are positive integers with \(m>n\), \(\gcd(m,n)=1\) and \(m\not\equiv n\pmod 2\). We show that if \(n=2\), then Jeśmanowicz’ conjecture is true. This is the first result that if \(n=2\), then the conjecture is true without any assumption on \(m\).

MSC:

11D61 Exponential Diophantine equations

Citations:

Zbl 0074.27205
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Full Text: DOI

References:

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