# 2021.11.15 MATH 3600 Consecutive prime Goldbach conjecture

## 194 days ago by calkin

In this worksheet we'll investigate generalizations of the Twin Prime Goldbach conjecture.  Let's start by considering consecutive primes which differ by exactly 4.

If $p-q=4$ then $p\equiv 5$(mod 6) and $q \equiv$1 (mod 6)

The primes p and q will be $3(2k+1) \pm 2$

We can define the 4-core for cousin primes, i.e. primes that differ by exactly 4, to be $(2k+1)$ where the primes are $3(2k+1) \pm 2$.

With this defniition, any sum of two 4-cores will of necessity be even: so we can ask, how often is each even number the sum of two 4-cores?

What do sums of pairs of primes at distance 6 look like?  How should we define the 6-core?

d=4 P=[] for i in srange(1000): if is_prime(i): P.append(i)
d_primes=[] for i in srange(1,len(P)): if P[i]-P[i-1]==d: d_primes.append([P[i-1],P[i]])
d_primes[:20]
 [[7, 11], [13, 17], [19, 23], [37, 41], [43, 47], [67, 71], [79, 83], [97, 101], [103, 107], [109, 113], [127, 131], [163, 167], [193, 197], [223, 227], [229, 233], [277, 281], [307, 311], [313, 317], [349, 353], [379, 383]] [[7, 11], [13, 17], [19, 23], [37, 41], [43, 47], [67, 71], [79, 83], [97, 101], [103, 107], [109, 113], [127, 131], [163, 167], [193, 197], [223, 227], [229, 233], [277, 281], [307, 311], [313, 317], [349, 353], [379, 383]]
d_cores=[] for i in srange(1,len(P)): if P[i]-P[i-1]==d: d_cores.append((P[i]-2)/3)
print(d_cores[:20])
 [3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127] [3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127]