## 19153Certain Prime Integers with an Elegant Property

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• Dec 2, 2007
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.