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A problem from Fermat's last theorem?
Consider the equation
a³ + b³ = c³
has no integer solutions according to Fermat (and Andrew Wiles)
can be proven easily if b = a? 2a³ = c³ and (³√ 2)a = c but ³√ 2 is irrational completes the proof.
now take case b≠a; say b = na where b>a without loss of generality so n>1;
get ³√ (n³ +1) = c/a means that there is no integer n where ³√ (n³ +1) is rational.
My question is try to find a number n>1 so that b³ - a³ = c³ exists for integers solution.
³√ (n³ -1) = c/a ... can this be done?
1 Answer
- PaulaLv 79 years agoFavorite Answer
No. Because:
b^3 - a^3 = c^3
=> b^3 = c^3 + a^3
... which has no positive integer solutions by Fermat's last theorem
(Need to specify positive integer, because otherwise you can set a=0)
