a) Can we establish a proof, there exists infinitely many primes of the form $n^2$ + 1. Why is the unit digit of such a prime p always 1 or 7? Is there any reasonable procedure or concept for the fact that the unit digit 7 occur essentially twice as often, when we identify the primes < 10000?
b) can we prove or disprove that, there exists an interval of the form [$n^2$, $(n+1)^2$] containing at least 1000 prime numbers.
c) We know that even integer > or = 4 can be written as sum of two primes and integers > 5 can be written as sum of three primes. Of course, those are conjectures. I am not asking the proof of those conjectures. I would like to know those statements are equivalent or not. If yes, how you will justify?