JM Bergot sent the following nice puzzle:
Is it uncommon it is for
three consecutive primes p,q,r to have p+q+r=3*q, as in
Contributions came from: Enoch Haga, Anton Vrba, Torbjörn Alm, Jan
van Delden, J. K. Andersen, Frederick Schneider, Jean Brette, Antoine
All of them pointed out an early mistake in the puzzle (p+q+r=3p was
wrong and we swiched to p+q+r=3q).
All of them pointed out that the asked primes are such that q-d, q,
q+d sum 3q.
Anton said that there are infinite solutions. Torbjörn said that
there are zillions of solutions CPAP type.
It is called a CPAP-3 (3 consecutive primes in arithmetic
The middle prime q is also called a balanced prime.
The 10 largest known cases are at
Below that table is a gigantic case with probable primes.
The largest known common difference is 21102 at
It is conjectured that there are infinitely many cases for every
common difference divisible by 6. But it has not even been proved
are infinitely many cases in total.
Through one million about 1 in 26 (2994 out of 78498) trios of
consecutive primes are in arithmetic progression (which is
to p+q+r=3*p). Through 10 million, about 1 in 30 (21837 out of
665677) meet this criterion. Through 100 million, about 1 in 34
(167031 out of 5761453 ) meet it Note the first trio is 3, 5, 7.
And so one in the same trend.