Unusual Sets of Consecutive Prime Numbers
- Here are 2 well known algebraic expressions; A=(((v-2)*r^2)-((v-4)*r))/2 and B=((v*(r^2))-v*r+2)/2. For positive integer values of v>2 and r>1, A evaluates to what is called a non-centered polygonal number, and B to a centered polygonal number. (v stands for vertices and r for rank)
I discovered that there seems to be a limitless number of sets of 4 positive odd consecutive primes P1<P2<P3<P4 that have a neat connection to the above expressions.
Take expression A and write six expressions by alternate substitution of each pair of adjacent primes in the set (P1<P2<P3<P4) for v and r thus: C1=(((P1-2)*P2^2)-((P1-4)*P2))/2, C2=(((P2-2)*P1^2)-((P2-4)*P1))/2, C3=(((P2-2)*P3^2)-((P2-4)*P3))/2, C4=(((P3-2)*P2^2)-((P3-4)*P2))/2, C5=(((P3-2)*P4^2)-((P3-4)*P4))/2, C6=(((P4-2)*P3^2)-((P4-4)*P3))/2. Look for the sets of 4 consecutive primes that upon evaluation of the 6 expressions; the 3 sums (C1+C2), (C2+C3) and (C3+C4) are simultaneously prime. You will find as many as you like.
The above statement also applies to expression B.
Remarkably, the 3 differences (gaps) between adjacent consecutive primes of the set always equal one of the terms in the arithmetical progression (6+4*n) where n=0, 1, 2, 3�n.
Here is the first occurrence of a set of 4 consecutive primes for expression A. P1=4591, P2=4597, P3=4603 and P4=4621. (C1+C2)=96892279097, (C2+C3)=97272496991 and (C3+C4)=98035582091. All sums are prime. The 3 gaps are 6, 6 and 18.
Here is the first occurrence of a set of 4 consecutive primes for expression B. P1=1078417, P2=1078471, P3=1078489 and P4=1078507. (C1+C2)= 1254273921685706303, (C2+C3)= 1254399534977724803 and (C3+C4)= 1254462344419147733. All sums are prime. The 3 gaps are 54, 18 and 18.
No luck so far with a set of 5. If there is one, it must really be huge. Is there some theory applicable to this exercise which explains why only primes with certain gaps work? Thanks folks.
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