4 ms·

> [1729] is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways. This isn’t actually correct. 1729 is

> [1729] is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways. 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).

psychoslave 4y ago
Is that still true with complex or higher dimension numbers in general? +832375400086541 if you can provide a proof through mathematical induction.
- renewiltord 4y ago
  There's no total order on complex numbers, so what would smallest mean there? The numbers z=x+iy (x,y in Z) with smallest positive |z|? Fair enough.
  - Y_Y 4y ago
    You make an important point, but I understood the parent as, "what is the smallest integer expressible as the sum of two cubes of Cayley-Dickson algebras of order n?" So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
  - ouid 4y ago
    >theres no total order on the complex numbers Sure there is, lex order on Re and Im and then inherit from the reals.
renewiltord 4y ago
> 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.
- yrdmb 4y ago
  > There is no value to this comment. 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.
  - Y_Y 4y ago
    > There is always value in truthful statements. 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?
- ttctciyf 4y ago
  > There is no value to this comment. I mostly just wanted to be annoying Since annoying provides negative value, your statement is incorrect.
  - renewiltord 4y ago
    Oh no! Hoist by my own petard of pedanticity! XD

psychoslave 4y ago

Is that still true with complex or higher dimension numbers in general? +832375400086541 if you can provide a proof through mathematical induction.

renewiltord 4y ago
There's no total order on complex numbers, so what would smallest mean there? The numbers z=x+iy (x,y in Z) with smallest positive |z|? Fair enough.
- Y_Y 4y ago
  You make an important point, but I understood the parent as, "what is the smallest integer expressible as the sum of two cubes of Cayley-Dickson algebras of order n?" So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
- ouid 4y ago
  >theres no total order on the complex numbers Sure there is, lex order on Re and Im and then inherit from the reals.

renewiltord 4y ago

> 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.

yrdmb 4y ago
> There is no value to this comment. 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.
- Y_Y 4y ago
  > There is always value in truthful statements. 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?
ttctciyf 4y ago
> There is no value to this comment. I mostly just wanted to be annoying Since annoying provides negative value, your statement is incorrect.
- renewiltord 4y ago
  Oh no! Hoist by my own petard of pedanticity! XD