A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triplets

Published on Jan 1, 2009in Acta Arithmetica0.421
· DOI :10.4064/AA138-2-3
Maohua Le1
Estimated H-index: 1
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2 Authors (Yondong Guo, Maohua Le)
1 Author (Maohua Le)
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#1Richard K. GuyH-Index: 23
This monograph contains discussions of hundreds of open questions, organized into 185 different topics. They represent aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In this second edition, new material ha...
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Let a,b,cbe relatively prime positive integers such that a^2+b^2=c^2, 2|b In this paper, we show that Pythagorean triples (a, b,c)must satisfy abc\equiv{0\; (\mod3\cdot{4}\cdot{5}})and c\neq{0\; (\mod{3}}) and we also prove that for (a,b,c)\in\{(a,b,c)|a\equiv{0\;(\mod{3}}),b\equiv{0\;(\mod{4}}),c\equiv{0\; (\mod{5}})\}\bigcup\{(a,b,c)|b\equiv{0\;(\mod{12}}),c\equiv{0\;(\mod{5}})\} the only solution of $a^x+b^y=c^z\qquad{z},y,z\in{N} in positive integers is x, y, z) = (2, ...
#1Bo HeH-Index: 8
#2Alain Togbé (Purdue University North Central)H-Index: 14
Last. Shichun YangH-Index: 3
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Click on the link to view the abstract. Keywords: Exponential Diophantine equation, Terai conjecture, positive integer solution, linear forms in two logarithms, lower bound, primitive divisor Quaestiones Mathematicae 36(2013), 119–135
#1Takafumi MiyazakiH-Index: 9
#2Pingzhi Yuan (SCNU: South China Normal University)H-Index: 16
Last. Danyao Wu (SCNU: South China Normal University)H-Index: 1
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Abstract In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple ( a , b , c ) satisfying a 2 + b 2 = c 2 , that the equation a x + b y = c z has the unique solution ( x , y , z ) = ( 2 , 2 , 2 ) in positive integers x , y and z . This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of classical well-known results on the conjecture. As a corollary we can verify that the conjecture is true if a − b = ± 1 .
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#1Zhi-Juan Yang (Anhui Normal University)H-Index: 2
#2Min Tang (Anhui Normal University)H-Index: 6
Let a,b,cbe relatively prime positive integers such that a^{2}+b^{2}=c^{2}.In 1956, Je\'{s}manowicz conjectured that for any given positive integer nthe only solution of (an)^{x}+(bn)^{y}=(cn)^{z}in positive integers is (x,y,z)=(2,2,2) In this paper, we show that (8n)^{x}+(15n)^{y}=(17n)^{z}has no solution other than (x,y,z)=(2,2,2)in positive integers. DOI: 10.1017/S000497271100342X
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