Can the Fermat's Last Theorem be expanded to higher powers?

[robert80]

You all know that the Fermats last theorem is solved for some years and that the equation

a^n + b^n = c^n

is solved when a,b,c being the natural numbers only for n = 2.

I would like to expand a problem:

Can anybody proove that:

a^n + b^n + c^n = d^n has a solutions a,b,c,d in the natural numbers for n = 3 and that for each higher n equation is non solveable?

Lets carry on: Can anybody proove that a^n + b^n + c^n + d^n + e^n = f^n for n = 4 the last solution exists?and for n>4 there are no solutions?

Thanks,

Robert

it would be very nice to find the rule, how many particles on the power of n you have to sum that you get the last solution of the equation in order of given n.

 

[D H]

You appear to be asking about Euler's conjecture, which is now known to be false.

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴
27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵

 

[robert80]

So you need at least n-1 natural numbers in order to define next one ? I don't know anything about Eulers conjecture...I don't even know it exists :)

 

[robert80]

Sorry now I see, haven't heard about it before. I have read it on Wikipedia now. But its funny that I got the idea without hearing f it :) Just some centuries too late :)

 

[robert80]

Just one more thing. Does this holds for any n? How can we know or assume, how many variables we need in order to describe a next one?Isnt it possible that when we reach n = 1000 we need much more variables (or less) than when n = 999? Does it have any importance in some special vector spaces? Thank you.

 

[D H]

Euler's conjecture was that solutions would only exist if the number of summands was greater than or equal to the power. His conjecture stated that, for example, integer solutions exist to a³+b³+c³=d³ and to a⁴+b⁴+c⁴+d⁴=e⁴ but not to a⁴+b⁴+c⁴=d⁴ or a⁵+b⁵+c⁵+d⁵=e⁵. The counterexamples in post #2 show that this is conjecture is not true.

You are leaping to conclusions based on those counterexamples. Don't do that.

 

[robert80]

Thank you so much, you are really kind. Yes I got that,,, BUT why don't the matematicians do the another theorem about powers of n and natural solutions in general? I mean, this Eulers conjecture is now proved to be wrong, so why don't they do better one, till is proved to be right or in worse case wrong? Those 2 are only specific cases for general Eulers Conjecture. And it is very true if you find the solution so, that for the lower n-s does not hold true. I think it should be proven to be wrong in general for every n.