Number theory

Peter Plichta's prime number cross

The numbers arranged in rings of 24. All primes from 5 on lie on just eight rays, which form a cross.

In the 1990s the chemist and private scholar Peter Plichta (born 1939) made the prime number cross known. Arrange the natural numbers in concentric rings of 24. Then all primes from 5 on lie on just eight of the 24 rays, namely those with remainder 1, 5, 7, 11, 13, 17, 19 or 23 on division by 24. These rays lie opposite each other in pairs, and together they form a cross.

The numbers 1 to 192 in rings of 24. Red: primes. Light red: numbers on the eight rays that are not prime, starting with 25 = 5 × 5.

That all primes greater than 3 lie on these eight rays follows directly from the rhythm of six: 24 = 4 × 6, and the eight remainders are exactly the numbers below 24 divisible neither by 2 nor by 3. But non-primes lie on the rays too, the first being 25, 35, 49 and 55. These are the same “pseudoprimes” described in The disorder of the primes.

Assessment

Plichta’s discovery and the importance he gave the number 24 deserve recognition. What can be criticised is that it has become a kind of doctrine that blanks out other approaches. When a retired engineer pointed Plichta to the grid of 30, which also contains all prime positions, he is said to have shown no interest.

The coordinate system under The tetractys and the Cartesian coordinate system is reminiscent of the prime number cross and deals with the same subject, but is built quite differently.

Why 24 in particular plays so many roles is summarised in The number 24.