Expended Fermats last theorem

  • Thread starter robert80
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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

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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
 
64
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So you need at least n-1 natural numbers in order to define next one ? I dont know anything about Eulers conjecture...I dont even know it exists :)
 
64
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Sorry now I see, havent 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 :)
 
64
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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

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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.
 
64
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Thank you so much, you are really kind. Yes I got that,,, BUT why dont 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 dont 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.
 

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