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

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The discussion centers on the equation a^n + b^n = c^n and whether it has real solutions as n approaches infinity. A proof is proposed that suggests no real solutions exist, using limits to argue that as n increases, the expression approaches zero rather than one. However, participants challenge the validity of discussing the equation at infinity, asserting that infinity is not a number and questioning the meaning of seeking solutions in that context. They emphasize that while solutions exist for finite values of n, the concept of n equaling infinity is mathematically nonsensical. Ultimately, the conversation highlights a misunderstanding of limits and the nature of mathematical expressions versus equations.
  • #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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