Is There a Real Solution for a^n+b^n=c^n as n Approaches Infinity?

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Discussion Overview

The discussion centers around the equation an + bn = cn and whether there are real solutions for a, b, and c as n approaches infinity. Participants explore the implications of limits in this context, questioning the validity of the original premise and the meaning of infinity in mathematical expressions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant attempts to prove that there are no real solutions for a, b, c as n approaches infinity, using limits to argue that the expression approaches 0 rather than 1.
  • Another participant questions the validity of the limit used in the proof, asking for clarification on its relevance.
  • Some participants assert that the question of solutions for n tending to infinity is meaningless, arguing that infinity is not a number and cannot be used in equations.
  • Others counter that while solutions exist for finite values of n, the inquiry into the behavior as n approaches infinity is still valid.
  • There is a discussion about the distinction between expressions and equations, with emphasis on the nature of limits and their interpretations in mathematical contexts.
  • One participant provides an example involving the Fibonacci sequence to illustrate the concept of limits, suggesting that limits can be meaningful in certain mathematical contexts.

Areas of Agreement / Disagreement

Participants express disagreement on the validity of discussing the equation as n approaches infinity. Some believe the question is nonsensical, while others argue that it is a legitimate inquiry into the behavior of the equation under certain conditions. No consensus is reached regarding the interpretation of limits and the implications for the original equation.

Contextual Notes

Participants highlight the limitations of using infinity in mathematical expressions, noting that it cannot be treated as a finite number. The discussion reveals a lack of clarity regarding the definitions and implications of limits in this context.

  • #61
matt grime said:
Both forms are not equivalent though because of that "limit" word in there (n is a dummy vairable, not actually a variable).

Yes, and this is the reason I had misgivings about solving equations with the limit notation in them. Certainly that works fine in special cases, as you illustrated.
 
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  • #62
this thread was so depressing from the beginning i never jumped in. questions need to be clearly stated before an answer is possible. the problem posed originally was never actually defined.

please stop this.
 
  • #63
If I could edit this, I would simply change "real" to "positive integer", and the problem would go away.
 

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