On Jeśmanowicz' conjecture concerning primitive Pythagorean triples

Terai, Nobuhiro

doi:10.1016/j.jnt.2014.02.009

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Introduction4

Conjecture 111

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2015

2022

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Cited by 35 publications

(15 citation statements)

References 12 publications

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“…After these works, Conjecture 1.1 has been proved to be true for various particular cases. For recent results, we only refer to the papers of the author [8], the author, Yuan and Wu [9], Terai [13], and the references given there.…”

Section: Conjecture 11mentioning

confidence: 99%

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Miyazaki

2015

Acta. Math. Sin.-English Ser.

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Section: Conjecture 11mentioning

confidence: 99%

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Miyazaki

2015

Acta. Math. Sin.-English Ser.

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“…Again, under the condition that 2 mn, Cao [1] proved the following two propositions. In 2014, Terai [16] proved that if n = 2, then Jeśmanowicz' conjecture is true without any assumption on m. Recently, Miyazaki and Terai [11] extended this result.…”

Section: Introductionmentioning

confidence: 96%

A Note on Jeśmanowicz’ Conjecture Concerning Primitive Pythagorean Triples

Deng

Huang

2016

Bull. Aust. Math. Soc.

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Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$, where $m$ and $n$ are positive integers with $m>n$, $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$. We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$. We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$. Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$.

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“…Jeśmanowicz [8] conjectured that if , , are Pythagorean numbers, i.e., positive integers satisfying 2 + 2 = 2 , then (1.1) has only the positive integer solution ( , , ) = (2, 2, 2) (cf. [14,17,22]). As an analogue of Jeśmanowicz' conjecture, the author proposed that if , , , , , are fixed positive integers satisfying + = with , , , , , ≥ 2 and gcd( , ) = 1, then (1.1) has only the positive integer solution ( , , ) = ( , , ) except for a handful of triples ( , , ) (cf.…”

Section: Introductionmentioning

confidence: 99%