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Dear Mr Wilson,

I was wondering if you could help solve an argument. The equation is x^3 - y^3 = 5803. I got an answer of x = 20, y = 13 and x = -13, y = -20. My friend says to solve this, just set y^3 = 0 and then the answer is y = 0, x = cube root 5803. There seems to be something wrong with his answer, but I can't show mathematically if he's right or wrong and why. Would you be able to help clarify things?

Sincerely,

AlexD

There is no unique solution to this.

Using Excel, an infinite number of solutions could be found.

As integers, you gave the only two that are correct.

See, 20^3 = 20*20*20 = 400*20 = 8,000 and 13^3 = 13*13*13 = 169 * 13 = 1690+507 = 2,197.

Thus, x^3 - y^3 = 8,000 - 2,197 = 5,803.

Allowing for any real numbers, he is correct as well.

Other answers for (x,y) could be

1 5804^(1/3)

2 5811^(1/3)

3 5830^(1/3)

4 5867^(1/3)

5 5928^(1/3)

6 6019^(1/3)

7 6146^(1/3)

8 6315^(1/3)

9 6532^(1/3)

10 6803^(1/3)

11 7134^(1/3)

12 7531^(1/3)

With a calculator, these can be found to be

1 17.97114719

2 17.97836908

3 17.99794215

4 18.03593644

5 18.09822841

6 18.19036643

7 18.31741502

8 18.48379394

9 18.6931314

10 18.94815042

11 19.25060436

12 19.60127022

The non-integer solutions are endless.

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