Answer to Question #184446 in Discrete Mathematics for Danish

Question

As you now that

0^{3 0}

1^{3 1}

2^{3 8}

3^{3 27}

4^{3 64}

5^{3 125}

so, we have a proof that every positive integer is the sum of the cubes of eight non negative integers. For example

7= 1³ + 1³ + 1³ +1³+1³+1³+ 1³ +0³

12= 2³+ 1³ + 1³ + 1³ +1³+0³+0³+0³

Can you disprove this statement?

Answer 1

Let's try to decompose 23 into the sum:

We need to use more 8 to minimize the number of summands:

$23=8+8+1+1+1+1+1+1+1$

There are nine summands so the statement is false