I discovered that certain primes have an elegant property. For

convenience, call this type of prime Q. I think I can best explain what I

mean by the following examples.

Take the prime Q=439. The largest prime P smaller than Q is 433 and the

smallest prime R larger than Q is 443.

Write Q in positional notation as (400+30+9). Get the 3 primes

immediately succeeding each of the terms (400, 30 and 9). They are (401,

31 and 11). Their sum is 443, which equals the above R. Get the 3 primes

immediately preceding each of the terms (400, 30 and 9). They are (397,

29 and 7). Their sum is 433, which equals the above P. Notice the twin

primes 29 and 31. As far as I went, at least one twin pair always showed

up with primes of this type.

The rule to follow in the above exercise is: If any term in the

positional notation is 0, then the succeeding prime is 2 and the

preceding prime is 0. If the term in the units position in the positional

notation is 1, then the succeeding prime is 2 and the preceding prime is

0.

And there are Q primes where only the sum of the succeeding primes equals

R. Here is an example of a chain of 4 Q's, namely (790733, 790739,

790747, 790753), where the sum of the succeeding primes of the first term

of the chain equals the second term, the sum of the succeeding primes of

the second term equals the third term, etc. So far I found no chain of 5

Q's.

A particularly interesting example is this set of 4 consecutive Q primes

in arithmetic progression: (23104127, 23104157, 23104187, 23104217). The

sum of the succeeding primes of the first term of the CPAP equals the

second term, the sum of the succeeding primes of the second term equals

the third term, etc. I found no web reference to CPAP's with this

property.

And there are plenty of examples of Q primes where only the sum of the

preceding primes equals P.

Thanks folks for your time and any comments. Hope it proves

recreationally interesting to someone.

Bill Sindelar