1. Twin Primes Conjecture : there are infinitely many pairs of Twin Primes among the infinitely many prime numbers.
Note: 2013 Zhang YiTang proved the gap between the twin primes is no more than 70,000,000 [Terence Tao’s Polymath Project using Zhang’s method to further narrow the gap to 246 ]
3. Palindromic Primes (eg.11, 101, 16561)