Problems & Puzzles: Puzzles

Puzzle 462. p+q+r=3*q 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 601+607+613=1821=3*607?

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Contributions came from: Enoch Haga, Anton Vrba, Torbjörn Alm, Jan van Delden, J. K. Andersen, Frederick Schneider, Jean Brette, Antoine Verroken.

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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.

Andersen wrote:

It is called a CPAP-3 (3 consecutive primes in arithmetic progression).
The middle prime q is also called a balanced prime.
The 10 largest known cases are at http://hjem.get2net.dk/jka/math/cpap.htm#k3
Below that table is a gigantic case with probable primes.
The largest known common difference is 21102 at http://hjem.get2net.dk/jka/math/cpap.htm#difference
It is conjectured that there are infinitely many cases for every common difference divisible by 6. But it has not even been proved that there
are infinitely many cases in total.

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Schneider wrote:

Through one million about 1 in 26 (2994 out of 78498) trios of
consecutive primes are in arithmetic progression (which is equivalent
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.

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And so one in the same trend.

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