Prime Spirals
by Simon Whitechapel
If 1 and 2 are counted as primes, any positive integer, n, can be represented as a sum of distinct primes, p1, p2, p3... To find the sum, subtract from n the prime, p1, nearest in size to n, then repeat for n1 = n0-p1; n2 = n1-p2; etc until nk = 0. When k, the number of primes in the sum, is 1, n is itself prime:
| n | p | k |
| 1 | = 1* | 1 |
| 2 | = 2 | 1 |
| 3 | = 3 | 1 |
| 4 | = 3 + 1* | 2 |
| 5 | = 5 | 1 |
| 6 | = 5 + 1 | 2 |
| 7 | = 7 | 1 |
| 8 | = 7 + 1 | 2 |
| 9 | = 7 + 2 | 2 |
| 27 | = 23 + 3 + 1* | 3 |
| 1354 | = 1327 + 23 + 3 + 1* | 4 |
When k = 1, n itself is prime; an asterized n sets a record for k (the n for k = 5 is very large). Here are Ulam spirals for k = 1..4.
k = 1
k = 2
k = 3
k = 4
© 2008 Simon Whitechapel
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