Prime and 24

The square of any prime number greater than 3 has an intriguing property: it is always one more than a multiple of 24. This characteristic can be a handy tool for preliminary checks when identifying prime numbers, although additional verification is still required.

To illustrate, consider the square of each prime number greater than 3, subtract 1 from it, and divide the result by 24:

Although this pattern can suggest primality, it is not definitive. For example, numbers like 25, 35, and 49 also fit this pattern, yet they are not primes.

To understand why this pattern holds, we can apply a bit of number theory using base 12. Any integer can be expressed in the form $12n + k$ where $k$ ranges from 0 to 11.

When we square $12n + k$ and subtract 1, the expression simplifies as follows:

$$\begin{split} (12n+k)^2 - 1 & = 144n^2 + 24kn + k^2 - 1 \\ & = 24n(6n + k) + (k^2 - 1) \end{split}$$

This expression is clearly divisible by 24, except for the term $k^2 - 1$, which requires specific conditions for divisibility by 24. Only certain values of $k$ associated with primes will satisfy this.

When investigating which values of $k$ may lead to prime numbers, most are disqualified by divisibility rules:

Testing the viable values of k yields:

Interestingly, 5, 7, 11 are indeed prime numbers, confirming the pattern.