I know a talented, enthusiastic, and very very hard-wroking 7th grade student, who began working on a research project about prime numbers a month ago. 

He has written numerous Pascal programs to find large prime numbers (using methods such as sieve of Eratosthenes) and twin primes, computing density of prime numbers and twin primes and their relative frequencies in successive intervals of arbitrary length, etc. His programs work quite fine and fast. He consistently improves his codes, e.g. his last code for finding prime numbers yields all primes less than 1000000 in 3 minutes (the first one took 3 hours!).

Also, he knows/has discovered some theoretical facts about prime numbers, e.g. how to prove that there are infinitely many primes, and why the twin primes are of the form $6k-1$ and $6k+1$.

**My Question:** What are the possible directions for him to continue his research project (via programming or pure mathematical practice), without facing technicalities?

Thanks.