This isn’t actually correct. 1729 is the smallest number expressible as a sum of two positive cubes in two different ways. The smallest number expressible as a sum of two cubes in two different ways is 91 (3^3+4^3 and -5^3+6^3).
That can't be true either since -91 = (-3)^3 + (-4)^3 and -91 = 5^3 + (-6)^3. Since -91 < 91, the statement has to be false.
In fact, it is not the number x with smallest value |x| that can be expressed as the sum of two cubes in two different ways since, 0 = (-1)^3 + 1^3 and 0 = (-2)^3 + 2^3.
It appears that 91 is the smallest positive integer that can be expressed as a sum of two cubes in two different ways.
There is no value to this comment. I mostly just wanted to be annoying. I hope I'm right, at least.
There is always value in truthful statements. You are right that 91 is not the smallest number as was implied by the parent. -91 is. After all it was the parent who brought in the negative integers into the mix.
As long as we're not talking about nonzero value.
I think GP made a worthwhile point, but in general the universe is full of uninteresting truths. Like take two large, distinct numbers, assert that they are unequal. Is that specific truth valuable?
Since annoying provides negative value, your statement is incorrect.
+832375400086541 if you can provide a proof through mathematical induction.
So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
Sure there is, lex order on Re and Im and then inherit from the reals.