SPSRs from orders 9 to 17 are sorted by percentage of primes in elements. o9-17spsr-prime.csv
format; "Order" "Width" "Height" "E1" ... "E17" (bouwkamp codes) "P1" ... "P17"(binary 1 if prime, 0 if not) "Prime Total" "Percent of elements prime"
The maximum proportion of primes among the elements of squared rectangles, up to order 17 is 64.71% (11/17) for a 260 x 258 rectangle. This is significantly higher than the proportion found in squared squares. Among squared squares, 3 examples are known of 50% prime. and one isomer pair of side 767 in order 29 which has 15 elements prime (51.7%) See primepuzzles.net.
There are actually 2 simple perfect squared squares (SPSSs) of 29 squares with more than 50% (51.7% = 15/29) of the squares prime, both SPSSs are 767 on the side, (which is not a prime), and have all the same pieces arranged differently (they are isomers), the different arrangement occurs in a small patch of the tiling.
These particular ones were discovered sometime over the last year by myself, Ed Pegg Jr and Stephen Johnson in our searches in order 29.
The bouwkampcodes are almost identical, just one transposition;
29 767 767 435 332 103 229 153 127 109 149 69 40 23 206
26 101 29 183 179 39 59 11 8 20 3 5 104 9 2 7 95
29 767 767 435 332 103 229 153 127 109 149 69 40 23 206
26 101 29 183 179 39 59 8 11 20 3 5 104 2 9 7 95
the 15 prime squares inside the 767 square are; 2, 3, 5, 7, 11, 23,
29, 59, 101, 103, 109, 127, 149, 179, 229
The interesting thing is the actual distribution of primes in SPSSs, if you count the number of primes in each squared square and sum over the whole of order 29, you get this table;
p , the number of primes in a SPSS; c(p), the number of SPSSs with p in that order.
| p | c(p) |
| 0 |
27 |
| 1 |
77 |
| 2 |
239 |
| 3 |
494 |
| 4 |
875 |
| 5 |
1150 |
| 6 |
1275 |
| 7 |
1208 |
| 8 |
1058 |
| 9 |
761 |
| 10 |
430 |
| 11 |
200 |
| 12 |
84 |
| 13 |
18 |
| 14 |
4 |
| 15 |
2 |
| >=16 |
0 |
At the top there are 27 SPSSs with no primes, and at the bottom end, theres the only 2 SPSSs with 15 primes (>50% prime).
If you graph the frequency counts in the last table, it looks like a normal distribution.
The only other place I've heard about primes and the normal distribution is the Erdos Kac Theorem - !