Solution to puzzle 61: Two cubes?

Let n be an integer.  Can both n + 3 and n² + 3 be perfect cubes?

---

If n + 3 and n² + 3 are both perfect cubes then their product must also be a perfect cube.

So consider (n + 3)(n² + 3) = n³ + 3n² + 3n + 9 = (n + 1)³ + 8.
This can be a perfect cube only if it is 8 more than another perfect cube, namely (n + 1)³.

The only pairs of perfect cubes that differ by 8 are (−8, 0) and (0, 8).
So we must have

• (n + 1)³ = (−2)³ n = −3;  or
• (n + 1)³ = 0³ n = −1.

For neither of these solutions is n² + 3 a perfect cube.

Therefore, if n is an integer, n + 3 and n² + 3 cannot both be perfect cubes.

---

One cube?

For what integers n is 18(n² + 3) a perfect cube?

Hint  -  Answer  -  Solution

Source: Inspired by Mathematical Miniatures, by Svetoslav Savchev and Titu Andreescu. See Coffee Break 1.

Back to top