Ulam spiral

The Ulam spiral (or Ulam cloth) consists of integers, starting from a given $n$ at the center, written in a spiral with the prime numbers highlighted or emphasized in some way. For example, writing a spiral with 42 at the center,

90	67	68	69	70	71	72
89	66	51	52	53	54	73
88	65	50	43	44	55	74
87	64	49	42	45	56	75
86	63	48	47	46	57	76
85	62	61	60	59	58	77
84	83	82	81	80	79	78

and then simply blanking the composites, we obtain

	67				71
89				53		73
			43
			47
		61		59
	83				79

Carrying on this process in more layers will show that most of the primes tend to fall on certain diagonals and not others.

This formation was first pondered by Stanisław Ulam with 1 at the center. He tried carrying out the process much further, and also tried different center values, but in each case the primes would cluster on certain diagonals and sparsely populate others. Because almost all primes are odd, it is easy to explain why they would tend to form diagonals, but much more difficult to explain why they fall on certain diagonals.

Title	Ulam spiral
Canonical name	UlamSpiral
Date of creation	2013-03-22 16:45:41
Last modified on	2013-03-22 16:45:41
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A41
Synonym	Ulam cloth
Synonym	Ulam spiral