Solutions to Exercises in the Introduction of How To Prove It by Daniel J Velleman.
Problem (6): The sequence 3, 5, 7 is a list of three prime numbers such that each pair of adjacent numbers in the list differ by two. Are there any more such “triplet primes”?
No. There are none. This can be trivially proven from the fact that in any given list of three consecutive odd numbers, one of them will always be divisible by 3. This means that there cannot exist any prime triplets other than 3, 5 and 7.