A student has asked me whether the following is true:
If p is prime and bigger than 37, there exists a three-term arithmetic progression of primes ending in p.
For smaller odd primes one can find 3-APs except for 3, 5, 13, and 37. For those the best one can do is (1,(p+1)/2,p). The next prime such that (p+1)/2 is prime is 61, but we can find a 3-AP (13,37,61).
(Edit) I have tested all primes up to 10^5 and found no other exceptions.
Any hints?