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): $6$