Prime twins and prime constellations : Tommy's conjecture

Tommy's conjecture
Let t(1)=1 , t(2)=3 and t(n) be the n'th triangular number.
Let all variables be integers >= 1.
Let ... mean that we do not care about that structure.
Use the OLD definition of primes that includes 1 as a prime then :
Between t(a)-1 and t(a+1)+1 always exists at least 1 prime.
Between t(a)-1 and t(a+3)+1 always exists at least 1 prime twin.
Between t(a)-1 and t(a+2k+1)+1 always exists at least 1 prime constellation of the form(p,...,p+2k).
This is the 'weak' version of my prime conjecture. There also exist stronger ones.

Below is a link to my profile at the tetration forum.

At the moment I only accept the tetration forum as a good source for tetration although you can find some info and links on wiki , I do however not accept the wiki page.


Thank you for your intrest.