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May 2nd, 2016, 12:52 PM   #1
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Untested Equation set. (Related to Fermat Last Theorem)

I was wondering if there's something similar to Fermat's Last Theorem but in this manner:

sum series of n terms each raised to the power of n being equal to another term raised to the same power

a1^n + a2^n + ... + an^n = b^n

1)
a^2 + b^2 = c^2
- - - - - - - - - - -
2)
a^3 + b^3 + c^3 = d^3
- - - - - - - - - - - - - - - -
3)
a^4 + b^4 + c+4 + d^4 = e^4


could it be that these all have no solutions above n?

so

1) no solution above 2
2) no solution above 3
3) no solution above 4

and so on?


i have not tried this and not sure where to begin testing for this but just thought i'd post this.

Last edited by Kensou77; May 2nd, 2016 at 12:54 PM.
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May 2nd, 2016, 08:38 PM   #2
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A Conjecture Of Euler.
Гипотеза Эйлера — Википедия
https://ru.wikipedia.org/wiki/%D0%93...B5%D1%80%D0%B0

Not true. In addition, some decisions extremely difficult.
For a cube there are solutions.
https://ru.wikipedia.org/wiki/%D0%97...B1%D0%B0%D1%85

I for example found such decisions.
number theory - Find all integer solutions to Diophantine equation $x^3+y^3+z^3=w^3$ - Mathematics Stack Exchange

For some solutions you can find there.
ag.algebraic geometry - Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$ - MathOverflow
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May 3rd, 2016, 12:57 PM   #3
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i mean for 3 terms, there is no solution for powers above 3
4 terms no solutions for powers above 4, etc

basically using the same principles of the fermat's last theorem but scaling it up.

or maybe the solution cap isn't simply >n (where n is number of terms) but rather a different value on a different scale?


so


1)
a^2 + b^2 = c^2
- - - - - - - - - - -
2)
a^3 + b^3 + c^3 = d^3
- - - - - - - - - - - - - - - -
3)
a^4 + b^4 + c+4 + d^4 = e^4



but these are invalid


1)
a^n + b^n = c^n
n>2
- - - - - - - - - - -
2)
a^n + b^n + c^n = d^n
n>3
- - - - - - - - - - - - - - - -
3)
a^n + b^n + c+n + d^n = e^n
n>4


OR instead of linear growth, maybe 2) will cap at a higher n and 3) will also cap at a much higher n

Last edited by Kensou77; May 3rd, 2016 at 12:59 PM.
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May 3rd, 2016, 08:33 PM   #4
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I said this hypothesis is not true.
Specifically you need each equation to solve.
This task is very difficult. Divination will not help here.
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May 4th, 2016, 07:04 AM   #5
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The hypothesis may be valid and deserving exploration if there are known examples of
a^3+b^3+c^3=d^3

a^4+b^4+c^4+d^4=e^4

etc.
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