Take the two prime gaps of $4*n+1$ and $4*n+3$ and the first ten million primes. Will we find that the number of gaps for each is about the same? One could see the possibilility of a prime gap race similar to that for primes of the form $4*n+1$ and $4*n+3$. Has anyone done this?
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1What do you mean by the two prime gaps? What are the two gaps when $n=10?$ say. – saulspatz Apr 04 '18 at 18:12
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If n =10, then the number of terms between the two primes is 10-1=9=4n+1. The difference between the primes is one thing, and the number of terms between them is an other. Take 17,19 and the prime gap is 40+1 and for 19,23 the prime gap is 4*0+3. – J. M. Bergot Apr 04 '18 at 18:57
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I got $$[4448503, 5551496]$$ The first entry is the number of gaps with residue $0$ modulo $4$ and the second the number of gaps with residue $2$ modulo $4$. Seems that residue $2$ modulo $4$ occurs significantly more often. – Peter Apr 04 '18 at 22:49
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I dare say, the rare courageous and bold getting the computer to produce some data! Odd that the second gives 6 while the first gives 12,24,36,48 and other multiples of 12. It appears that the second with 6, 18,30,42...is enough to be a winner. – J. M. Bergot Apr 05 '18 at 18:19