Why is it that (if you exclude 2 & 3) the difference between the squares of any two prime numbers is divisible by 12?

Physicist: That’s a really cool property!

Every prime number (other than 2 and 3) can be written in the form 6j+1 or 6j+5. For example, 17 = 6(2)+5 and 31 = 6(5)+1.

This is because numbers of the form 6j, 6j+2, 6j+4, are all divisible by 2, and all numbers of the form 6j, 6j+3 are divisible by 3. So that restricts the options for primes to just the “+1″ set and the “+5″ set. Not all of the numbers in these sets are primes, but all the primes are in these two sets.

6j = 6, 12, 18, 24, 30, … All divisible by 6.

6j+1 = 7, 13, 19, 25, 31, …

6j+2 = 8, 14, 20, 26, 32, … All divisible by 2.

6j+3 = 9, 15, 21, 27, 33, … All divisible by 3.

6j+4 = 4, 10, 16, 22, 28, 34, … All divisible by 2.

6j+5 = 5, 11, 17, 23, 29, 35, …

Call the primes p and q, and notice that .

If p and q are both the same type (+1 or +5), then (p-q) will be a multiple of 6. For example: (+1 case) 31-7 = 24 and (+5 case) 29-11 = 18.

If p and q are opposite types, then (p+q) will be divisible by 6. For example: 23+13 = 36.

In both cases, the other bubble, (p+q) or (p-q), will always be divisible by 2, since the sum and difference of any two odd numbers is always even. So, one bubble is always a multiple of 6 and the other is always a multiple of 2, and together the whole thing is always a multiple of 12.

For example: p=11, q=7. 18 is divisible by 6, and 4 is divisible by 2, so 18×4 is divisible by 12!

This is another example of modular arithmetic. It almost should have been included in the “tricks with 9′s post“.

Also: This trick doesn’t really have much to do with “primes”, so much as it has to do with “numbers that don’t have 2 or 3 as a factor”. That isn’t obvious at first. The first composite (not prime and not 1) number with no 2′s or 3′s is 25.



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