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]