Diophantine Equation with no solutions

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Discussion Overview

The discussion revolves around the solvability of a specific Diophantine equation of the form C^u = A^u + B^s, particularly for the case where u = 2s. Participants explore whether integer solutions exist for various values of s, especially focusing on the implications of Fermat's Last Theorem.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if the equation C^u = A^u + B^s has no integer solutions for all values of s.
  • Another participant notes that the equation C^2 = A^2 + B has integer solutions for s = 1.
  • A follow-up inquiry asks whether there are no solutions for s > 1 and if a proof exists for this assertion.
  • Concerns are raised about the nature of the restrictions, with one participant emphasizing the difference between integers and natural numbers greater than 2.
  • One participant suggests that a solution to C^22 = A^22 + B^11 would imply a solution to D^11 = E^11 + B^11, which they argue is impossible according to Fermat's Last Theorem, applying this reasoning to all positive integers s apart from 1.
  • A later reply reiterates the impossibility of the derived equation per Fermat's Last Theorem, indicating agreement with the previous point.

Areas of Agreement / Disagreement

Participants express differing views on the existence of solutions for s > 1, with some arguing for impossibility based on Fermat's Last Theorem while others seek clarification on the conditions and restrictions involved.

Contextual Notes

Participants discuss the implications of different types of integers (e.g., positive integers vs. natural numbers) and the potential need for a complete problem description, indicating that assumptions and definitions may influence the discussion.

e2m2a
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Does anyone know if this Diophantine equation is impossible to solve for all values of s?

C^u = A^u + B^s, where u = 2s

For example, is there no integer solution for C,A, and B for the following:

C^22 = A^22 + B^11
 
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It is possible to solve ##C^2 = A^2 + B ## with integers for s = 1
 
But what about for s > 1? Are there no solutions and if so, is there a proof for this?
 
Are the restrictions going to come in one at a time or is there a complete problem description ?
'Integers' is quite a bit more than 'natural numbers > 2' ##\qquad ## [edit] ##\ge 2##
 
Last edited:
BvU said:
Are the restrictions going to come in one at a time or is there a complete problem description ?
'Integers' is quite a bit more than 'natural numbers > 2'
Interested only in positive integers or natural numbers for s > 1,
 
A solution to C22 = A22 + B11 would be a solution to D11 = E11 + B11 where D = C2 and E = A2, which is impossible per Fermat's last theorem. Similar for all other positive integers s apart from 1.
 
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mfb said:
A solution to C22 = A22 + B11 would be a solution to D11 = E11 + B11 where D = C2 and E = A2, which is impossible per Fermat's last theorem. Similar for all other positive integers s apart from 1.
Ok. That makes sense. Thanks.
 

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