#math trivia for #February28:#59 and 61 are twin primes. What property of 60 makes this possible? (Hint:29 and 31 were also twins.)
— Burt Kaliski Jr. (@modulomathy) February 28, 2012
The property of 60 that makes adjacent twin primes possible is that 60 is divisible by 6.
If 60 were not divisible by 2, then both of its neighbors would have to be, so neither could be a prime.
If 60 were not divisible by 3, then one of its neighbors would have to be, so the pair of neighbors would not be primes.
It also helps a little that 60 is divisible by 5. If 60 were not divisible by 5, then there would be a 50% probability that one of its neighbors would be divisible by 5, and for that reason not prime. (Consider the first four numbers divisible by 6 but not 5: 6, 12, 18, 24. Two have neighbors divisible by 5, and two don’t.)
So, being neighbors of a number by 6 is a necessity for twin primes, and being neighbors of a number divisible by 30 or 60 is even better.