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 experts ignore Plichta’s prime cross and his thesis of the key role of 24. Anyone who has not read his books should be careful with judgements. Anyone who reads them, though, can guess why: some derivations by analogy do not survive scrutiny, interpretations like those under Mysticism and Philosophy on this site do not belong in a specialist book, and private and professional matters are constantly mixed, in a tone that puts off many readers. That should not tempt one to dismiss the factual content as well. Plichta is above all a chemist and looks for correspondences between the prime cross and the structure of atoms. This is not entirely far-fetched, as the link between the Riemann hypothesis and physics shows, which mathematicians and physicists now discuss actively (see Prime distribution – a matter of perspective).
Why 24?
The cross gives no answer to why the primes become irregular right after 24. The division into 24 still makes sense:
- Up to 24 the twin primes are ordered without a gap.
- Within the sections of 24 there is a prime coding: for every prime p above 3, p² − 1 is divisible by 24.
- Within the sections of 24 there is also a decimal coding.
| p | p² − 1 | as a multiple of 24 |
|---|---|---|
| 5 | 24 | 1 · 24 |
| 7 | 48 | 2 · 24 |
| 11 | 120 | 5 · 24 |
| 13 | 168 | 7 · 24 |
| 17 | 288 | 12 · 24 |
| 19 | 360 | 15 · 24 |
The reason: p − 1 and p + 1 are two consecutive even numbers, one of them divisible by 4, and one of the three numbers p − 1, p, p + 1 is divisible by 3, but not p.
Behind the cross lies a simple order. Divide the natural numbers into three groups by their remainder on division by three: the groups with remainder 1 and remainder 2 jump in pairs, the numbers divisible by three stand still and are evenly spread. Because this order is mirrored, the rhythm of six arises. That is only the tip of the iceberg, though, or perhaps the foundation. Plichta’s thoughts on the rhythm of six after Leibniz and on 24 are taken up in The order of the primes, as is the question of whether 1 is a prime and the special role of 2 and 3. Philolaus already distinguished two proper forms of number, odd and even, and a third mixed from both, the even-odd, each with many shapes (see The ancient sources).
One might think this order depends only on our decimal system and that other orders hold in other counting systems. That is wrong. It rests on a strict rotation matrix that coincides completely with the divisor properties of the simplex figures, and there is only one geometry of simplexes, not a separate one for each counting system (see Simplex – the solid form of number). Whether one sees in this a subtlety or a universal principle, everyone must decide for themselves after checking it.
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.