Question on Fermat's Last Theorem

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Discussion Overview

The discussion revolves around Fermat's Last Theorem and its implications regarding powers and their relationships, particularly focusing on the nature of 6th powers, squares, and Pythagorean triples. Participants explore theoretical connections and mathematical reasoning related to these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that according to Fermat's Last Theorem, a 6th power plus a 6th power cannot equal a 6th power, while a square plus a square can equal a square.
  • There is a suggestion that 6th powers can be expressed as squares of 3rd powers, and that any even powers can be written as squares.
  • One participant questions whether this reasoning could help identify numbers that cannot form Pythagorean triples, suggesting a relationship between powers and integer solutions.
  • Another participant asserts that the known Pythagorean triples do not require Fermat's Last Theorem for their identification.
  • Further elaboration is provided on the conditions under which pairs of squares can sum to a square, with references to specific forms of integers and their relationships to cubes.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Fermat's Last Theorem and its relevance to Pythagorean triples. There is no consensus on the connections being drawn between these concepts, and the discussion remains unresolved regarding the broader implications of the theorem.

Contextual Notes

Some statements rely on specific mathematical definitions and assumptions that may not be universally accepted. The discussion includes unresolved mathematical reasoning and does not clarify the conditions under which certain relationships hold.

Wiz14
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According to Fermat's last theorem, a 6th power plus a 6th power cannot equal a 6th power, but a square plus a square can equal a square. But can't 6th powers be written as squares of 3rd powers? Also, can't any even powers be written as squares?
 
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Wiz14 said:
But can't 6th powers be written as squares of 3rd powers? Also, can't any even powers be written as squares?

Yes, they can be written as such. Not really sure what you're getting at...
 
micromass said:
Yes, they can be written as such. Not really sure what you're getting at...

Is this a way to know what numbers cannot be pythagorean triples? for example:
a2 + b2 = c2
cannot have integer solutions if a,b, and c are perfect squares, cubes, fourth powers, etc?
 
Wiz14 said:
Is this a way to know what numbers cannot be pythagorean triples?

We know what all the pythagorean triples are. You don't need FLT for this.
 
Wiz14 said:
According to Fermat's last theorem, a 6th power plus a 6th power cannot equal a 6th power, but a square plus a square can equal a square. But can't 6th powers be written as squares of 3rd powers? Also, can't any even powers be written as squares?
Well yes even powers can be written as squares, but only certain pairs of squares sum to a square. FLT which has been proven states that x^y + y^n = z^n has no solution in integers. Thus, for n > 2 including n equal to an even number greater than 2 there is no solution.

P.S. it has been shown that all solutions of the form x^2 + y^2 = z^2 are such that the pair x and y are of the form a^2 - b^2 and 2ab for some a and b, but a^2 - b^2 can never equal a cube except in the case |a| = 3 and |b| = 1. However, in that case 2ab = +/- 6 which is not a cube. Thus 8^2 + 6^2 = 10^2 has the property that 8 is cube. Although all even cubes are of the form 2ab, no two cubes sum to a square except in the trival cases where the square is 0 or one of the cubes is 0, i.e. a=b=k^3 or either of (a,b) = 0 in which case the common requirement that a is coprime to b is missing.
 
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