# Consecutive Integers which are Powers of 2 or 3

## Contents

## Theorem

The only pairs of consecutive positive integers which are powers of $2$ or $3$ are:

- $\tuple {1, 2}$, $\tuple {2, 3}$, $\tuple {3, 4}$, $\tuple {8, 9}$

## Proof

Let $a$ and $b$ be two arbitrary consecutive positive integers.

### Both Integers Powers of $2$

The only powers of $2$ that differ by $1$ are $2^0$ and $2^1$, which gives us the pair $\tuple {1, 2}$.

### Both Integers Powers of $3$

The powers of $3$ are:

- $1, 3, 9, \ldots$

and so trivially there are no two consecutive positive integers which are powers of $3$.

### The Remaining Case

There is no need to consider the case where $a$ or $b$ is $1$, as that has already been investigated.

So, let $a = 2^m$ and $b = 3^n$ where both $m$ and $n$ are greater than $0$.

From Powers of 3 Modulo 8, $b \equiv 1 \pmod 8$ or $b \equiv 3 \pmod 8$.

For $m \ge 3$ we have that:

- $2^m \equiv 0 \pmod 8$

while:

- $2^1 \equiv 2 \pmod 8$
- $2^2 \equiv 4 \pmod 8$

Let $a = b + 1$.

Then we have the cases:

- $b = 1, a = 2$

which has already been covered, and:

- $b = 3, a = 4$

which gives us the pairs $\tuple {3, 4}$.

Otherwise, for $a \ge 8$, we have:

\(\displaystyle b + 1\) | \(=\) | \(\displaystyle a\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b + 1\) | \(\equiv\) | \(\displaystyle 0\) | \(\displaystyle \pmod 8\) | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b\) | \(\equiv\) | \(\displaystyle 7\) | \(\displaystyle \pmod 8\) |

and we have demonstrated that there are no such $b$, as $b \equiv 1 \pmod 8$ or $b \equiv 3 \pmod 8$

Let $b = a + 1$.

Let $b = 3^n$ where $n$ is odd.

Then $b \equiv 3 \pmod 8$

which means:

- $b \equiv 2 \pmod 8$

leading to the pair $\tuple {2, 3}$.

No further pairs are possible for $b = 3^n$ with $n$ odd.

Let $b = 3^n$ where $n$ is even.

Thus:

\(\displaystyle a\) | \(=\) | \(\displaystyle 2^m\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle b - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3^n - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3^{2 k} - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {3^k + 1} \paren {3^k - 1}\) | Difference of Two Squares | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {3^k + 1} \paren {3^k - 1}\) | Difference of Two Squares | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^m\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^p \, 2^q\) | for some $p, q \in \Z_{\ge 0}$ such that $2^p = 2^q + 2$ |

The only such $p$ and $q$ are $2^p = 2$ and $2^q = 4$

This means that the only possible pair fulfilling this condition is $\tuple {8, 9}$.

Thus all such possible pairs have been found.

$\blacksquare$

## Historical Note

The result **Consecutive Integers which are Powers of 2 or 3** was demonstrated rigorously by Levi ben Gershon, under the name Leo Hebraeus, in his *De numeris harmonicis* of $1343$.