# An application of the Baker method to Jeśmanowicz’ conjecture on Pythagorean triples

An application of the Baker method to Jeśmanowicz’ conjecture on Pythagorean triples Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple with \$\$a^2+b^2=c^2\$\$ a 2 + b 2 = c 2 . A positive integer solution (x, y, z) of the equation \$\$(an)^x+(bn)^y=(cn)^z\$\$ ( a n ) x + ( b n ) y = ( c n ) z is called exceptional if \$\$(x,y,z)\ne (2,2,2)\$\$ ( x , y , z ) ≠ ( 2 , 2 , 2 ) . Sixty years ago, L. Jeśmanowicz conjectured that, for any n, the equation has no exceptional solutions. This problem is not resolved as yet. In this paper, using the Baker method, we prove that if \$\$n>1\$\$ n > 1 , \$\$b+1=c\$\$ b + 1 = c and \$\$c>500000\$\$ c > 500000 , then the equation has no exceptional solutions (x, y, z) with \$\$y>z>x\$\$ y > z > x . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Springer Journals

# An application of the Baker method to Jeśmanowicz’ conjecture on Pythagorean triples

, Volume 112 (2) – Feb 28, 2017
6 pages

/lp/springer_journal/an-application-of-the-baker-method-to-je-manowicz-conjecture-on-7H3JSvEn8L
Publisher
Springer Milan
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Theoretical, Mathematical and Computational Physics
ISSN
1578-7303
eISSN
1579-1505
D.O.I.
10.1007/s13398-017-0384-9
Publisher site
See Article on Publisher Site

### Abstract

Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple with \$\$a^2+b^2=c^2\$\$ a 2 + b 2 = c 2 . A positive integer solution (x, y, z) of the equation \$\$(an)^x+(bn)^y=(cn)^z\$\$ ( a n ) x + ( b n ) y = ( c n ) z is called exceptional if \$\$(x,y,z)\ne (2,2,2)\$\$ ( x , y , z ) ≠ ( 2 , 2 , 2 ) . Sixty years ago, L. Jeśmanowicz conjectured that, for any n, the equation has no exceptional solutions. This problem is not resolved as yet. In this paper, using the Baker method, we prove that if \$\$n>1\$\$ n > 1 , \$\$b+1=c\$\$ b + 1 = c and \$\$c>500000\$\$ c > 500000 , then the equation has no exceptional solutions (x, y, z) with \$\$y>z>x\$\$ y > z > x .

### Journal

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. MatemáticasSpringer Journals

Published: Feb 28, 2017

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