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Wikipedia says Fermat's last theorem has the greatest number of failed proofs in history. I presume this simple "proof" is one of them. It must have been thought up before me. I first considered it years ago when I first heard of the problem. Figured it was so simple someone else must have thought of it before and disposed of it so I didn't really flesh it out then. My question is for anyone familiar with the history of the theorem is why this "proof" wouldn't be considered an acceptable proof.
First consideration is that all Pythagorean triples with a, b, or c greater than the square root of 2 are scale able by the same coefficient for a b and c that leaves c equal to the square root of 2. This disposes of all numbers greater than the square root of 2 for a b and c.
Second consideration is that when a=1 b=1 and c=sqrt2. As a approaches 0 b approaches sqrt2. Trivially when a=0 b=sqrt2 and c=sqrt2. As 0 isn't a positive number it isn't included in fermat's last theorem because Fermat says the theorem covers positive values for a b and c.
Third consideration is when a =1 then a to any exponent n =1 and b to n=1 and c (=sqrt2) to any n larger than 2 is going to be greater than 2 and therefore cannot be a to n+b to n=c to n when n>2.
Fourth consideration; as the first consideration says c=sqrt 2 divided by any scaling coefficient when c> sqrt2 then a to n + b to n must equal 2 else a squared + b squared would not equal c squared. So as a to n approaches 0 b to n approaches but is always smaller than c to n.
Therefore any a to n + b to n cannot equal c to n for any number a b and c or any n larger than 2. That's Fermats' last theorem in a nutshell.
So why is this "proof" not satisfactory.
If you have any questions about the above, something not clear, I could flesh it out some more for you. If you know if this "proof" has been previously published and disposed of I'd appreciate a direction to look into the disposition of this.
First consideration is that all Pythagorean triples with a, b, or c greater than the square root of 2 are scale able by the same coefficient for a b and c that leaves c equal to the square root of 2. This disposes of all numbers greater than the square root of 2 for a b and c.
Second consideration is that when a=1 b=1 and c=sqrt2. As a approaches 0 b approaches sqrt2. Trivially when a=0 b=sqrt2 and c=sqrt2. As 0 isn't a positive number it isn't included in fermat's last theorem because Fermat says the theorem covers positive values for a b and c.
Third consideration is when a =1 then a to any exponent n =1 and b to n=1 and c (=sqrt2) to any n larger than 2 is going to be greater than 2 and therefore cannot be a to n+b to n=c to n when n>2.
Fourth consideration; as the first consideration says c=sqrt 2 divided by any scaling coefficient when c> sqrt2 then a to n + b to n must equal 2 else a squared + b squared would not equal c squared. So as a to n approaches 0 b to n approaches but is always smaller than c to n.
Therefore any a to n + b to n cannot equal c to n for any number a b and c or any n larger than 2. That's Fermats' last theorem in a nutshell.
So why is this "proof" not satisfactory.
If you have any questions about the above, something not clear, I could flesh it out some more for you. If you know if this "proof" has been previously published and disposed of I'd appreciate a direction to look into the disposition of this.
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