Ulam spiral

The Ulam spiral is an arrangement of numbers upon which the prime numbers tend to fall on certain diagonals. It may begin on any integer, and may be oriented a few different ways. But the version most commonly presented begins with 1 at the center, and takes its cue for orientation from the March 1964 cover of Scientific American, with 2 to the right of 1, and 3 above 2, so that the northern ray reads 1, 4, 15, 34...

Original Ulam spiral

In the spiral, even numbers are shown in red, odd numbers in black and prime numbers are in bold.

A053755: Numbers of the form $4n^{2}+1$ A121326: Primes of this form	A054556: Numbers of the form $4n^{2}-9n+6$ A168023: Non-composites of this form					A002378: Numbers of the form $n(n+1)$ 2 is the only even prime				A054554: Numbers of the form $4n^{2}-10n+7$ A073337: Primes of this form
A054567: Numbers of the form $4n^{2}-7n+4$ A168025: Non-composites of this form	65	64	63	62	61	60	59	58	57	A054552: Numbers of the form $4n^{2}-3n+1$ A168022: Non-composites of this form
	66	37	36	35	34	33	32	31	56
	67	38	17	16	15	14	13	30	55
	68	39	18	5	4	3	12	29	54
	69	40	19	6	1	2	11	28	53
	70	41	20	7	8	9	10	27	52
	71	42	21	22	23	24	25	26	51
	72	43	44	45	46	47	48	49	50
	73	74	75	76	77	78	79	80	81
A054569: Numbers of the form $4n^{2}-6n+3$ A168026: Non-composites of this form	A033951: Numbers of the form $4n^{2}+3n+1$ A168027: Non-composites of this form					A073577: Numbers of the form $(2n+1)^{2}-2$ A028871: Primes of this form				A016754: Odd squares, $(2n+1)^{2}$ No primes of this form.

Stanislaw Ulam first studied this spiral in 1963, doodling "while sitting through a boring talk." Other starting values can be used and the same clustering of primes along certain diagonals is also observed, such as starting with 41.

Odd numbers Ulam spiral

In the odd number variant of the Ulam spiral, unimpeded by the even numbers, the prime numbers can line up in horizontal and vertical lines. But there are still noticeable diagonal lines of primes, and these primes fall on one such diagonal.

221	223	225	227	229	231	233	235	237	239	241
219	145	147	149	151	153	155	157	159	161	163
217	143	85	87	89	91	93	95	97	99	165
215	141	83	41	43	45	47	49	51	101	167
213	139	81	39	13	15	17	19	53	103	169
211	137	79	37	11	1	3	21	55	105	171
209	135	77	35	9	7	5	23	57	107	173
207	133	75	33	31	29	27	25	59	109	175
205	131	73	71	69	67	65	63	61	111	177
203	129	127	125	123	121	119	117	115	113	179
201	199	197	195	193	191	189	187	185	183	181

Note the diagonal of primes of the form $\scriptstyle 8n^{2}-1\,$ (Cf. A090684)

{7, 31, 71, 127, 199, 647, 967, 1151, 1567, 2311, 2591, 2887, 3527, 4231, 4999, 5407, 6271, 7687, 8191, 11551, 12799, 16927, 19207, 20807, 23327, 25087, 27847, 31751, 34847, 35911, ...}

Note the horizontal line of primes of the form $\scriptstyle 8n^{2}+2n+1\,$ (Cf. A188382)

{11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, 2927, 3571, 3917, 4657, 5051, 6329, 8779, 9871, 11027, 14197, 14879, 17021, 20101, 21737, 26107, 27967, 28921, 33931, 34981, ...}

Note the vertical line of primes of the form $\scriptstyle 8n^{2}+6n-1\,$ (Cf. A187677)

{13, 43, 89, 151, 229, 433, 701, 859, 1033, 1223, 1429, 1889, 2143, 2699, 3001, 3319, 4003, 4751, 5563, 7873, 10009, 11173, 11779, 12401, 13693, 17203, 18719, 19501, 21943, 25423, ...}

Ulam spiral with numbers congruent to 1 or 5 (mod 6)

331	335	337	341	343	347	349	353	355	359	361
329	217	221	223	227	229	233	235	239	241	245
325	215	127	131	133	137	139	143	145	149	247
323	211	125	61	65	67	71	73	77	151	251
319	209	121	59	19	23	25	29	79	155	253
317	205	119	55	17	1	5	31	83	157	257
313	203	115	53	13	11	7	35	85	161	259
311	199	113	49	47	43	41	37	89	163	263
307	197	109	107	103	101	97	95	91	167	265
305	193	191	187	185	181	179	175	173	169	269
301	299	295	293	289	287	283	281	277	275	271

External links

• From BackIssues.com, cover of the Scientific American March 1964 issue
• YouTube, Ulam spiral video with digits
• YouTube, Ulam spiral video without digits