Spirals of Prime Numbers

Stanislaw Ulam ID badge photo from Los Alamos

Public Domain, https://commons.wikimedia.org/w/index.php?curid=315600

Instructions

Begin with any number (limited to 0 and positive integers: 1, 2, 3, 4…) in the center of the spiral. Continue writing down successive numbers until several layers of the spiral have been built up. You generally need at least 50-60 numbers, but more is better for observing the pattern. Use the grids below to create your spirals.

Example 1

Starting number: 0

					4	3	2
					5	0	1
					6

 

Continue adding numbers to the spiral:

		64	63	62	61	60	59	58	57	56
		65	36	35	34	33	32	31	30	55
		66	37	16	15	14	13	12	29	54
		67	38	17	4	3	2	11	28	53
		68	39	18	5	0	1	10	27	52
		69	40	19	6	7	8	9	26	51
		70	41	20	21	22	23	24	25	50
		71	42	43	44	45	46	47	48	49
		72	73	74	75	76	77	78	79	80

 

Next, highlight or circle the prime numbers. For your reference the prime numbers under 200 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

		64	63	62	61	60	59	58	57	56
		65	36	35	34	33	32	31	30	55
		66	37	16	15	14	13	12	29	54
		67	38	17	4	3	2	11	28	53
		68	39	18	5	0	1	10	27	52
		69	40	19	6	7	8	9	26	51
		70	41	20	21	22	23	24	25	50
		71	42	43	44	45	46	47	48	49
		72	73	74	75	76	77	78	79	80

 

A pattern appears:

Regardless of the starting number in the middle, many of the prime numbers – but not all – will form diagonal lines in the grid. In the example above, observe the diagonal that goes from near the upper left corner down to the lower right formed by the primes: 37, 17, 5, 7, 23, 47, 79. Also, the other major diagonal that includes the primes: 71, 41, 19, 5, 3, 13, 31.

Exercises:

Create your own spirals starting with any numbers. One particularly interesting example is the spiral that begins with 17. It has a diagonal 14 primes in a row.

For parents, teachers and advanced thinkers:

The formal name of the spiral pattern is called the Spiral of Ulam, named after the Polish-American mathematician, Stanislaw Ulam (1909-1984). Ulam worked on the Manhattan Project during World War II.

Background on Prime Numbers

A prime number is any number greater than 1 that only can be divided evenly by 1 or itself. The first two prime numbers are 2 and 3. Each can only be divided by 1 or itself. The number 4 is not prime because it can be divided evenly by 2 as well as by 1 and itself. That is, 1 x 4 = 4; 2 x 2 = 4; and 4 x 1 = 4. The next prime numbers are 5 and 7. The number 9 is not prime because 3 x 3 =9. It has been proven that the list of primes goes on without end. While there is no way to predict which numbers will be prime, there are some general rules which state which numbers cannot be prime. For example, even numbers greater than 2 cannot be prime because each even number greater than two will have 2 as a divisor. Also, numbers greater than 5 that end with a 5 or a 0 (such as 10, 15, 20, 25…) cannot be prime because they are all divisible by 5.

Ulam Spiral Google Sheets Link

Image of large Ulam Spiral (red dots are prime; blue: primes of form  )

(image credits: Will Orrick)

$$
{\displaystyle 4n^{2}-2n+41}

Please see this other post for another spiral of prime numbers.