Let t(1)=1 , t(2)=3 and t(n) be the n'th triangular number.
Let all variables be integers >= 1.
Between t(a)-1 and t(a+1)+1 always exists at least 1 prime.
Similarly let s(1) = 1, s(2) = 4 , ... be the n' th square number.
Between s(a)-1 and s(a+1)+1 always exists at least 2 primes.
Let p(n) be the n'th pentagonal number then
Between p(a)-1 and p(a+1)+1 always exists at least 2 primes.
This can be generalized
a polygonal number is a number that counts dots arranged in the shape of a regular polygon.
If s is the number of sides in a polygon, the formula for the nth s-gonal number P(s,n) is
P(s,1) = 1
P(s,2) = s
and in general
P(s,n) = [ (s-2) n^2 - (s-4) n ] / 2 = (s-3) t(n-1) + t(n)
define pi(s) where pi(s) is the number of primes from 2 up to s.
Then we have
Tommy's prime gap conjecture :
Between P(s,a) - 1 and P(s,a+1) + 1 always exist at least pi(s) primes.