What is the smallest positive integer that is the sum of 2 distinct perfect cubes in 2 different ways?

Note: A perfect cube is a cube of an integer (which could be positive, negative, or zero).

The answer is 91.

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@Brian Charlesworth I was equally surprised too. I was looking at the Fermat-Pythagoream Problem and realized interestingly that $c$ could be negative.

Chung Kevin
- 4 years, 8 months ago

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We also have that $9^{3} - 1^{3} = 6^{3} + 8^{3} = 12^{3} - 10^{3} = 728$ , so there are at least two values less than the Taxicab number.

Brian Charlesworth
- 4 years, 8 months ago

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Oh wow! Sum of cubes in 3 different ways! Maybe that could lead to another problem?

I like how you manipulated the Taxicab number. That leads to another example like $12 ^ 3 - 9 ^3 = 10^3 - 1 ^3 = 999$ :)

Chung Kevin
- 4 years, 8 months ago

Can we not prove it mathematically??

rajdeep brahma
- 3 years ago

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The only way that I know of is trial and error for all smaller values.

I don't know how to show that it is the smallest, other than listing out all the values and checking.

Chung Kevin
- 3 years ago

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oh ok I see thanks!!!!pls share if u get any proof on this pls

rajdeep brahma
- 3 years ago

Actually I disagree the answer, because I have found a much smaller number: $19$ .

It's $3^{3}-2^{3}$ and equivalent to $2.5^{3}+1.5^{3}$

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I edited in "perfect cubes". Otherwise, we could have $1 = \sqrt[3]{2} ^ 3 - 1^3 = \sqrt[3]{9} ^ 3 - 2^3$ .

Chung Kevin
- 3 years, 2 months ago

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I think it should say perfect cubes of integers or something similar.

We can extend the ring of integers by the roots of the polynomial $(x^3 - 2)(x^3 - 9) = x^6 - 11x^3 + 18$ to get a ring of algebraic integers containing both $\sqrt[3]{2}$ and $\sqrt[3]{9}$ .

In that ring it is perfectly valid to say that $2$ and $9$ are perfect cubes, yielding the solution $1$ you described.

Of course, perfect cube usually means either a cube of a positive integer or a cube of a negative integer if not otherwise stated, but this ambiguity led me to first try the famous $1729$ .

Jesse Nieminen
- 2 years, 10 months ago

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I added a comment to the question.

Even though I get what you mean, it becomes very smart-aleky. It's like saying

Well, 5 is not a prime under

Guassianintegers. So, you have to define which integer system you're in when you say "primes in the integers".

Chung Kevin
- 2 years, 10 months ago

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@Chung Kevin – Sorry. My intention wasn't to be cocky. I just get annoyed when the answer I've given should be correct, but is regarded as incorrect because of a mistake or, in this case, an ambiguity in the problem.

Jesse Nieminen
- 2 years, 10 months ago

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Most people would think of the taxicab number $10^3 + 9^3 = 1729 = 12^3 + 1 ^3$ .

This is the smallest positive integer that is the sum of 2 distinct

positivecubes in 2 different ways.However, because we are allowed to use negative cubes, there is a smaller answer, which is

$3^3 + 4^3 = 91 = 6^3 + (- 5)^3.$

I don't know how to show that it is the smallest, other than listing out all the values and checking.

In the comments, other examples which are smaller than the taxicab number are:

$9^{3} + (- 1)^{3} = 6^{3} + 8^{3} = 12^{3} + (- 10)^{3} = 728$ ,

$12 ^ 3 + (- 9) ^3 = 10^3 + (- 1) ^3 = 999$