# Difference between 2 Consecutive Cubes is Odd

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## Theorem

Let $a$ and $b$ be consecutive integers.

Then $b^3 - a^3$ is odd.

## Proof 1

Let $a, b \in \Z$ such that $b = a + 1$.

Then:

\(\ds b^3 - a^3\) | \(=\) | \(\ds \paren {a + 1}^3 - a^3\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds a^3 + 3 a^2 + 3 a + 1 - a^3\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 3 a \paren {a + 1} + 1\) |

From Product of Consecutive Integers is Even, $a \paren {a + 1}$ is even.

Hence $3 a \paren {a + 1} + 1$ is odd.

Hence the result.

$\blacksquare$

## Proof 2

Let $a, b \in \Z$ such that $b = a + 1$.

Either:

or:

Hence from Parity of Integer equals Parity of Positive Power either:

or:

The result follows.

$\blacksquare$

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $9$