# Prime equals Plus or Minus One modulo 6

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## Theorem

Let $p$ be a prime number greater than $3$.

Then $p$ is either of the form:

- $p = 6 n + 1$

or:

- $p = 6 n - 1$

That is:

- $p = \pm 1 \pmod 6$

## Proof

To demonstrate that there are prime numbers of either form, note:

- $5 = 6 \times 1 - 1$
- $7 = 6 \times 1 + 1$

The only other possibilities for $p$ are:

- $p = 6 n$, in which case $6 \divides p$ and so $p$ is not prime
- $p = 6 n + 2$, in which case $2 \divides p$ and so $p$ is not prime
- $p = 6 n + 3$, in which case $3 \divides p$ and so $p$ is not prime
- $p = 6 n + 4$, in which case $2 \divides p$ and so $p$ is not prime
- $p = 6 n + 5$, which is the same as $p = 6 \paren {n + 1} - 1$, which is covered above.

Hence the result.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$