It's very known that 1729 is the smallest number that can be expressed as the sum of two cubes in two different ways, but is there a proof that not uses brute force?
If we let $n$ be the smallest number such that $n=a^3+b^3=c^3+d^3$ with $a\neq c$, $b \neq d$ and $a,b,c,d > 0$, then:
$n = (a+b)(a^2-ab+b^2)$
$n=(c+d)(c^2-cd+d^2)$
But I don't have any idea how to follow from here. If we take a look at 1729:
1729=13·19·7
It seems obvious that if a number is the product of only 2 primes then it can't be expressed as the sum of two cubes in two different ways, but I don't know how to prove either that the least number which such property must be product of 3 primes or more.
Does anyone has any idea of how to approach this?