This simple fact blows up into a bunch of even more surprising things that are true of all primes when you follow the algebra to its logical conclusions. Like:
Since this means every pair of twin primes > 3 must be separated by a multiple of 6, so they can be written as 6k+1, 6k-1. That means the product of any pair of twin primes will be of the form 36k^2-1.
In other words take any pair of twin primes, multiply them together, add 1, divide by 36, you are guaranteed to get a perfect square. E.g. 11*13=143, +1=144, /36=4 =2^2.
Or (and this one’s actually a little more complicated because it gets kind of casewise) you can show that the square of any prime (>3) is either one more or one less than a multiple of 24 (which is the product of the 6 and the 4 from the 6k and 4k rules)
Since this means every pair of twin primes > 3 must be separated by a multiple of 6, so they can be written as 6k+1, 6k-1. That means the product of any pair of twin primes will be of the form 36k^2-1.
In other words take any pair of twin primes, multiply them together, add 1, divide by 36, you are guaranteed to get a perfect square. E.g. 11*13=143, +1=144, /36=4 =2^2.
Or (and this one’s actually a little more complicated because it gets kind of casewise) you can show that the square of any prime (>3) is either one more or one less than a multiple of 24 (which is the product of the 6 and the 4 from the 6k and 4k rules)