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March 19th, 2018, 10:11 PM   #1
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Fermat's Last for >3 terms, n>0

Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Is there a solution for four or more such terms with an integer n>0?
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March 20th, 2018, 04:20 AM   #2
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$\displaystyle 40^3+17^3+2^3=41^3$
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March 20th, 2018, 12:32 PM   #3
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I meant to write "is there always a solution to four or more such terms for each integer n>0?"
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March 20th, 2018, 02:04 PM   #4
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$3^3+4^3+5^3=6^3$
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March 20th, 2018, 06:07 PM   #5
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In general, does there always exist at least one solution for given n where

a^n+b^n+c^n+ ... +d^n=z^n

with [at least] n+1 terms, having a, b, c, d, z and n integers greater than zero?

__________

This is an extension of Fermat's Last Theorem.
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March 21st, 2018, 01:30 PM   #6
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Quote:
Originally Posted by Loren View Post
In general, does there always exist at least one solution for given n where

a^n+b^n+c^n+ ... +d^n=z^n

with [at least] n+1 terms, having a, b, c, d, z and n integers greater than zero?

__________

This is an extension of Fermat's Last Theorem.
Why "at least"? Why not less?
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March 21st, 2018, 04:59 PM   #7
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Including "at least" would have put fewer constraints on my equation. My guess is, that would make a less defined equation with "looser" results. The bracket showed my initial uncertainty. But here, without that bracketed phrase, I am looking for possible unique results for any given n>3.

It has been shown previously that there are multiple cases where n=3. My equation has a minimum of n=1 terms separated by an equal sign. "Less than" n+1 terms, where n=0, would not indicate an equation.

Fermat's Last Theorem eliminated all possible n=2 for its 3 terms, with the exception of n=1 and n=2. I included the symmetry I did for my hypothesis, so n is fundamentally related to the number of terms. mathman, your suggestion is worthwhile; I was trying to narrow the field of answers by considering a greater symmetry for n>3, with perhaps singular answers for each n exponents with their n+1 terms.

__________

Does there always exist at least one solution for given n where

a^n+b^n+c^n+ ... +d^n=z^n

with n+1 terms, having a, b, c, d, z and n integers greater than zero?
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March 24th, 2018, 01:53 PM   #8
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I posted this question on another forum, where you should get more responses.

https://math.stackexchange.com/
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March 25th, 2018, 10:48 AM   #9
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Much appreciation for the site, mathman. How do I find my post there?
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March 25th, 2018, 01:26 PM   #10
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At the end of the list you can click on popular tags. Next page click on users. The search for my name (herb steinberg). You will see the title "Generalized Fermat's Last Theorem".
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