# Fermat's Last Theorem: implications (there is no new proof)

I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number Theory leading to the development of new methods which can be themselves applied not only for the proof of FLT (like financial problems motivated somehow development of stochastic analysis). I am interested - if there are applications or implications of FLT itself?

More precisely: if the fact "for each $n\geq3$ there are no integer solutions of $a^n+b^n=c^n$" leads to solutions of problems which are not in the field of Number Theory?

I would specify that I wonder about some problems which are already formulated: since FLT is known for more then 300 years I am pretty sure that there were formulated hypothesis which follow from FLT directly (if there are such hypothesis not in the field of Number Theory).

## 2 Answers

Actually, FLT has very few consequences even in Number Theory. Its importance has always been in the methods that were developed in the effort to settle it, and what else could be done with those methods.

• The Riemann hypothesis is quite different. Many results on number theory is based on the "solution" of Riemann hypothesis (common belief is the hypothesis is true.) – Sunni Jun 26 '11 at 13:13

I've read a number theory paper (on Heron Triangles) that used FLT. See Proposition 14 of this paper.

• I don't think this example is very compelling since it only uses the solution of the Diophantine equation $a^2 = x^2 + y^2$. – Eric M. Schmidt Aug 17 '17 at 18:04

## protected by Zev ChonolesMar 6 '13 at 10:49

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?