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

Loading next page...
 
/lp/springer_journal/an-application-of-the-baker-method-to-je-manowicz-conjecture-on-7H3JSvEn8L
Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Italia
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off