A Note on Je{\'s}manowicz' Conjecture for Non-primitive Pythagorean Triples

Published on Feb 22, 2021in arXiv: Number Theory
Van Thien Nguyen1
Estimated H-index: 1
,
Viet Kh. Nguyen1
Estimated H-index: 1
,
Pham Hung Quy1
Estimated H-index: 1
Sources
Abstract
Let (a, b, c)be a primitive Pythagorean triple parameterized as a=u^2-v^2,\ b=2uv,\ c=u^2+v^2\ where u>v>0are co-prime and not of the same parity. In 1956, L. Je{\'s}manowicz conjectured that for any positive integer n the Diophantine equation (an)^x+(bn)^y=(cn)^zhas only the positive integer solution (x,y,z)=(2,2,2) In this connection we call a positive integer solution (x,y,z)\ne (2,2,2)with n>1exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case v=2,\ uis an odd prime. As an application we show the truth of the Je{\'s}manowicz conjecture for all prime values u < 100
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References16
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#1Maohua LeH-Index: 1
#2Reese ScottH-Index: 4
Last. Robert StyerH-Index: 4
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Let a b cbe fixed coprime positive integers with \min\{a,b,c\}>1 In this survey, we consider some unsolved problems and related works concerning the positive integer solutions (x,y,z)of the ternary purely exponential diophantine equation a^x + b^y = c^z
#1Mou Jie Deng (Haida: Hainan University)H-Index: 3
#2Dong Ming Huang (Haida: Hainan University)H-Index: 1
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#1Hai Yang (Xi'an Polytechnic University)H-Index: 2
#2Ruiqin Fu (Xi'an Shiyou University)H-Index: 1
Abstract Let ( a , b , c ) be a primitive Pythagorean triple such that a = u 2 − v 2 , b = 2 u v , c = u 2 + v 2 , where u , v are positive integers satisfying u > v , gcd ⁡ ( u , v ) = 1 and 2 | u v . In 1956, L. Jeśmanowicz conjectured that the equation ( a n ) x + ( b n ) y = ( c n ) z has only the positive integer solutions ( x , y , z , n ) = ( 2 , 2 , 2 , m ) , where m is an arbitrary positive integer. A positive integer solution ( x , y , z , n ) of the equation is called exceptional if (...
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Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2. Jeśmanowicz’ conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN) x + (bN) y = (cN) z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.
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#1Nobuhiro Terai (Ashikaga Institute of Technology)H-Index: 6
Abstract In 1956, Jeśmanowicz conjectured that the exponential Diophantine equation ( m 2 − n 2 ) x + ( 2 m n ) 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 ≢ n ( mod 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 .
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Let \((a, b, c)\) be a primitive Pythagorean triple satisfying \(a^2 +b^2 = c^2.\) In 1956, Je\'smanowicz conjectured that for any given positive integer \(n\) the only solution of \((an)^x + (bn)^y = (cn)^z\) in positive integers is \(x = y = z = 2.\) In this paper, for the primitive Pythagorean triple \((a, b, c)= (4k^2 - 1, 4k , 4k^2 + 1)\) with \(k=2^s\) for some positive integer \(s\geq 0\), we prove the conjecture when \(n >1\) and certain divisibility conditions are satisfied. 10.1017/S00...
Let a , b , c be relatively prime positive integers such that a 2 + b 2 = c 2 with b even. In 1956 Jeśmanowicz conjectured that the equation a x + b y = c z has no solution other than ( x , y , z )=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b . The main results of this paper include the case where 8 divides b . One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can b...
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#1Michael A. Bennett (UBC: University of British Columbia)H-Index: 19
#2Christopher Skinner (UM: University of Michigan)H-Index: 18
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subse- quently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.
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#1Moujie DengH-Index: 1
#2Graeme L. Cohen (UTS: University of Technology, Sydney)H-Index: 9
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 = ( en ) z in positive integers is x = y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > 1, c = b + 1 and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples ( a , b...
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#1Maohua LeH-Index: 1
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