Show no solutions exist for $x^3 + y^3 = 7777781.$

Question

For the following equation $$x^3 + y^3 = 7777781$$ we have to prove no solutions exist.

How to go about this? Will it require a modular arithmetic property?

Answer 1

For $x,y \in \mathbb{Z}$ you can see that $x^3 \equiv 0,1,-1 (\bmod 7) \implies x^3 +y^3 \equiv 0,1, -1, 2, -2 (\bmod 7.)$ However, $7777781 \equiv 4 (\bmod 7)$. Therefore there is no integer solution.

Answer 2

There is a complete characterisation of integers $n$ which are a sum of two cubes, see the article Characterizing the Sum of Two Cubes by Kevin A. Broughan. Several modular constraints are given, for example the following:

Let $n\in \Bbb N$ be a positive integer satisfying one of the congruences below. Then $n=x^3 +y^3$ has no solution in $\Bbb Z$:

$(1)$ We have $n\equiv 3,4 \bmod 7$.

$(2)$ We have $n\equiv 3,4,5,6 \bmod 9$

$(3)$ We have $n\equiv 3,4,5,6,10,11,12,13,14,15,17,18,21, 22, 23, 24, 25, 30, 31, 32, 33, 38, 39, 40, 41, 42, 45, 46, 48, 49, 50, 51, 52, 53, 57, 58, 59,60 \bmod 63$