Prime squares and square squares
Hello Math-Fun,
today we'll look for "prime squares" and "square squares".
Definition 1
A "prime square" is a prime which is the sum of 4 integers. Those integers must occupy the 4 vertices of a geometric square formed by 4 cell-centers properly chosen on an infinite grid A (the 4 integers hereunder in the upper left corner, for instance, sum up to 11, which is prime).
The type-A infinite grid is an infinite sector of the plane where the cells are labeled by the successive anti-diagonals like below:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
today we'll look for "prime squares" and "square squares".
Definition 1
A "prime square" is a prime which is the sum of 4 integers. Those integers must occupy the 4 vertices of a geometric square formed by 4 cell-centers properly chosen on an infinite grid A (the 4 integers hereunder in the upper left corner, for instance, sum up to 11, which is prime).
The type-A infinite grid is an infinite sector of the plane where the cells are labeled by the successive anti-diagonals like below:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+.
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|
+
|
Definition 2
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+.
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|
+
|
Definition 2
A "square square" is a square which is the sum of 4 integers. Those integers occupy the 4 vertices of a geometric square formed by 4 cell-centers properly chosen on an infinite grid B (the four integers 6, 1, 4 and 5 hereunder sum up to 16, for instance, which is a square).
The type-B grid is produced starting somewhere in the plane with 1 and labeling the successive cells clockwise — like the spiral-path hereunder:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
The type-B grid is produced starting somewhere in the plane with 1 and labeling the successive cells clockwise — like the spiral-path hereunder:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110|
73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109|
72| 43| 44| 45| 46| 47| 48| 49| 50|
83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108|
71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107|
70| 41| 20| 7 | 8 | 9 | 10|
27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106|
69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105|
68| 39| 18| 5 | 4 | 3 | 12|
29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104|
67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103|
66| 37| 36| 35| 34| 33| 32| 31| 56|
89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102|
65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100|
99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
An example of prime square on the A grid is [1 + 3 + 5 + 2] = 11
(the geometric square has side-length = 1):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
Another example of prime square on the A grid is [3 + 14 + 18 + 4] = 29
(the geometric square has side-length = SQR5)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A third example of prime square on the A grid is [1 + 10 + 25 + 7] = 43
(the geometric square has side-length = 3)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A 4th example of prime square on the A grid is [2 + 14 + 32 + 11] = 59
(the geometric square has side-length = 3 again)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A 5th example of prime square on the A grid is [13 + 52 + 84 + 30] = 179
(the geometric square has side-length = SQR17)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
As it is impossible to find any prime square on a type-B grid, we will look instead for square squares like [6 + 1 + 4 + 5] = 16 .
(The blue geometric square here has side-length = 1):
Another example of square square on the B grid is [46 + 24 + 8 + 22] = 100
(the geometric square has side-length = SQR2):
A 3rd example of square square on the B grid is [1 + 12 + 27 + 24] = 64
(thank you Scott!) (the geometric square has side-length = SQR5):
A 4th example of square square on the B grid is [1 + 86 + 127 + 116] = 324
(thank you again Scott!) (the geometric square has side-length = 5):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A 5th example of square square on the B grid is [1 + 41 + 67 + 35] = 144
(thank you again Scott!) (the geometric square side-length = SQR10):
Note that no odd square (like 121, 169 or 2025) is constructible on the above spiral-grid (B-type).
And now the question time:
(1) Can you illustrate all geometric side-lengths of the prime squares? [above, in green, we have the side-lengths 1, SQR5, 3 (twice) and SQR17, for instance].
(2) Can you find all the square squares constructible on a type-B grid?
Remarks:
In the same spirit, we could look for "triangular squares" on both grids (The first triangular numbers are visible here, in the OEIS:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176,...)
The idea expressed in this page came from three sources:
— Werner Keym's last book
— This recent video by Numberphile on YouTube
— This sequence of the OEIS ("numbers that are the sum of two squares"):
0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160,...
Best,
É.
____________________
Update, July 3rd, 2020
Scott R. Shannon was quick to compute a few (!) square squares on the B-grid (the "spiral") – here is what he e-mailed me:
(the geometric square has side-length = 1):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
Another example of prime square on the A grid is [3 + 14 + 18 + 4] = 29
(the geometric square has side-length = SQR5)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A third example of prime square on the A grid is [1 + 10 + 25 + 7] = 43
(the geometric square has side-length = 3)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
(the geometric square has side-length = 3 again)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A 5th example of prime square on the A grid is [13 + 52 + 84 + 30] = 179
(the geometric square has side-length = SQR17)
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 37| 47| 58| 70| 83| 97|112| | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 46| 57| 69| 82| 96|111| | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 56| 68| 81| 95|110| | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 67| 80| 94|109| | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 79| 93|108| | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 92|107| | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|106| | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
As it is impossible to find any prime square on a type-B grid, we will look instead for square squares like [6 + 1 + 4 + 5] = 16 .
(The blue geometric square here has side-length = 1):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
Another example of square square on the B grid is [46 + 24 + 8 + 22] = 100
(the geometric square has side-length = SQR2):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+A 3rd example of square square on the B grid is [1 + 12 + 27 + 24] = 64
(thank you Scott!) (the geometric square has side-length = SQR5):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+A 4th example of square square on the B grid is [1 + 86 + 127 + 116] = 324
(thank you again Scott!) (the geometric square has side-length = 5):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
A 5th example of square square on the B grid is [1 + 41 + 67 + 35] = 144
(thank you again Scott!) (the geometric square side-length = SQR10):
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |157|158|159|160|161|162|163|164|165|166|167|168|169| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |156|111|112|113|114|115|116|117|118|119|120|121|122| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | |145|144|143|142|141|140|139|138|137|136|135|134|133| | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+Note that no odd square (like 121, 169 or 2025) is constructible on the above spiral-grid (B-type).
And now the question time:
(1) Can you illustrate all geometric side-lengths of the prime squares? [above, in green, we have the side-lengths 1, SQR5, 3 (twice) and SQR17, for instance].
(2) Can you find all the square squares constructible on a type-B grid?
Remarks:
In the same spirit, we could look for "triangular squares" on both grids (The first triangular numbers are visible here, in the OEIS:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176,...)
The idea expressed in this page came from three sources:
— Werner Keym's last book
— This recent video by Numberphile on YouTube
— This sequence of the OEIS ("numbers that are the sum of two squares"):
0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160,...
Best,
É.
____________________
Update, July 3rd, 2020
Scott R. Shannon was quick to compute a few (!) square squares on the B-grid (the "spiral") – here is what he e-mailed me:
> Below are results for 'square square' , with 1
always in corner, for x and y offset up to 5000. Note that because the x offset
and the y offset can be up to 5000 it means the size of the top of the square
can be up to sqrt(2)*5000. Also note that my spiral goes anticlockwise so you
will need to flip top to bottom when looking for these numbers on the spiral
drawn on your webpage.
Startcorner = 1 xOff
= 2 yOff = -1 1:12:27:24 -> 64
Startcorner = 1
xOff = 3 yOff = 3 1:49:163:43 -> 256 (diagonal)
Startcorner = 1
xOff = 5 yOff = 0 1:86:121:116 -> 324
Startcorner = 1
xOff = 12 yOff = 6 1:535:1357:607 -> 2500
Startcorner = 1
xOff = 13 yOff = 5 1:633:1359:711 -> 2704
Startcorner = 1
xOff = 16 yOff = -16 1:993:4001:1089 -> 6084
Startcorner = 1
xOff = 20 yOff = -10 1:1551:3501:1671 -> 6724
Startcorner = 1
xOff = 23 yOff = -7 1:2055:3495:2193 -> 7744
Startcorner = 1 xOff
= 31 yOff = -1 1:3753:3971:3939 -> 11664
Startcorner = 1
xOff = 32 yOff = 32 1:4225:16577:4161 -> 24964 (diagonal)
Startcorner = 1
xOff = 45 yOff = 3 1:7963:9403:8233 -> 25600
Startcorner = 1
xOff = 51 yOff = -27 1:10279:24079:10585 -> 44944
Startcorner = 1
xOff = 54 yOff = 33 1:11470:30559:11794 -> 53824
Startcorner = 1
xOff = 60 yOff = -18 1:14239:24061:14599 -> 52900
Startcorner = 1
xOff = 61 yOff = 59 1:14643:57963:15009 -> 87616
Startcorner = 1
xOff = 62 yOff = -55 1:15246:54399:15618 -> 85264
Startcorner = 1
xOff = 69 yOff = 18 1:18820:30589:19234 -> 68644
Startcorner = 1
xOff = 70 yOff = -43 1:19434:50711:19854 -> 90000
Startcorner = 1
xOff = 72 yOff = 0 1:20521:21025:20953 -> 62500
Startcorner = 1
xOff = 74 yOff = -31 1:21714:43743:22158 -> 87616
Startcorner = 1
xOff = 75 yOff = -30 1:22306:43741:22756 -> 88804
Startcorner = 1 xOff
= 78 yOff = 69 1:24034:86887:24502 -> 135424
Startcorner = 1
xOff = 85 yOff = 50 1:28596:73341:29106 -> 131044
Startcorner = 1
xOff = 91 yOff = -22 1:32874:50669:33420 -> 116964
Startcorner = 1
xOff = 99 yOff = -90 1:38998:142309:39592 -> 220900
Startcorner = 1
xOff = 105 yOff = 57 1:43729:105511:44359 -> 193600
Startcorner = 1
xOff = 106 yOff = 29 1:44598:73383:45234 -> 163216
Startcorner = 1
xOff = 107 yOff = -10 1:45486:54309:46128 -> 145924
Startcorner = 1
xOff = 112 yOff = 8 1:49833:58065:50505 -> 158404
Startcorner = 1
xOff = 122 yOff = 53 1:59118:123095:59850 -> 242064
Startcorner = 1
xOff = 124 yOff = 38 1:61095:105549:61839 -> 228484
Startcorner = 1
xOff = 125 yOff = 50 1:62076:123101:62826 -> 248004
Startcorner = 1
xOff = 126 yOff = 63 1:63064:143515:63820 -> 270400
Startcorner = 1
xOff = 127 yOff = -121 1:64257:245267:65019 -> 374544
Startcorner = 1
xOff = 128 yOff = 84 1:65069:180457:65837 -> 311364
Startcorner = 1
xOff = 133 yOff = 14 1:70344:86997:71142 -> 228484
Startcorner = 1
xOff = 140 yOff = -118 1:78099:265461:78939 -> 422500
Startcorner = 1
xOff = 141 yOff = 48 1:79054:143545:79900 -> 302500
Startcorner = 1
xOff = 152 yOff = -100 1:92061:253209:92973 -> 438244
Startcorner = 1
xOff = 157 yOff = 80 1:98046:225465:98988 -> 422500
Startcorner = 1
xOff = 158 yOff = -157 1:99540:395955:100488 -> 595984
Startcorner = 1
xOff = 163 yOff = -102 1:105890:280045:106868 -> 492804
Startcorner = 1
xOff = 163 yOff = -94 1:105882:263357:106860 -> 476100
Startcorner = 1
xOff = 164 yOff = 138 1:106955:365749:107939 -> 580644
Startcorner = 1
xOff = 173 yOff = 39 1:119159:180547:120197 -> 419904
Startcorner = 1
xOff = 174 yOff = -15 1:120598:142159:121642 -> 384400
Startcorner = 1
xOff = 178 yOff = 59 1:126144:225507:127212 -> 478864
Startcorner = 1
xOff = 181 yOff = 158 1:130344:460725:131430 -> 722500
Startcorner = 1
xOff = 189 yOff = 174 1:142144:528181:143278 -> 813604
Startcorner = 1
xOff = 190 yOff = -93 1:143924:319411:145064 -> 608400
Startcorner = 1
xOff = 195 yOff = -192 1:151708:597913:152878 -> 902500
Startcorner = 1
xOff = 197 yOff = 6 1:154640:165637:155822 -> 476100
Startcorner = 1
xOff = 202 yOff = -55 1:162666:263279:163878 -> 589824
Startcorner = 1
xOff = 206 yOff = 5 1:169122:178919:170358 -> 518400
Startcorner = 1
xOff = 207 yOff = -45 1:170821:253099:172063 -> 595984
Startcorner = 1
xOff = 211 yOff = -72 1:177524:319369:178790 -> 675684
Startcorner = 1
xOff = 214 yOff = -51 1:182594:279943:183878 -> 646416
Startcorner = 1
xOff = 221 yOff = 143 1:194559:531155:195885 -> 921600
Startcorner = 1
xOff = 231 yOff = -27 1:212779:265279:214165 -> 692224
Startcorner = 1
xOff = 232 yOff = -16 1:214617:245057:216009 -> 675684
Startcorner = 1
xOff = 234 yOff = 159 1:218164:619051:219568 -> 1056784
Startcorner = 1 xOff
= 243 yOff = -207 1:235675:808615:237133 -> 1281424
Startcorner = 1
xOff = 250 yOff = -205 1:249456:826691:250956 -> 1327104
Startcorner = 1
xOff = 266 yOff = -181 1:282408:797811:284004 -> 1364224
Startcorner = 1
xOff = 269 yOff = 33 1:288605:365959:290219 -> 944784
Startcorner = 1
xOff = 282 yOff = 231 1:317020:1054267:318712 -> 1690000
Startcorner = 1
xOff = 283 yOff = -144 1:319652:727897:321350 -> 1368900
Startcorner = 1
xOff = 292 yOff = 158 1:340023:811485:341775 -> 1493284
Startcorner = 1
xOff = 296 yOff = 68 1:349509:531305:351285 -> 1232100
Startcorner = 1
xOff = 299 yOff = -16 1:356724:395673:358518 -> 1110916
Startcorner = 1
xOff = 306 yOff = 33 1:373594:460975:375430 -> 1210000
Startcorner = 1
xOff = 317 yOff = 138 1:400868:829645:402770 -> 1633284
Startcorner = 1
xOff = 319 yOff = 319 1:408321:1630091:407683 -> 2446096 (diagonal)
Startcorner = 1
xOff = 322 yOff = -105 1:413876:727819:415808 -> 1557504
Startcorner = 1
xOff = 323 yOff = -10 1:416358:442245:418296 -> 1276900
Startcorner = 1
xOff = 325 yOff = 68 1:421458:619233:423408 -> 1464100
Startcorner = 1
xOff = 326 yOff = 129 1:423998:829663:425954 -> 1679616
Startcorner = 1
xOff = 333 yOff = 210 1:442348:1181149:444346 -> 2067844
Startcorner = 1
xOff = 334 yOff = 29 1:445194:528471:447198 -> 1420864
Startcorner = 1
xOff = 339 yOff = -225 1:458893:1270579:460927 -> 2190400
Startcorner = 1
xOff = 345 yOff = 105 1:474961:811591:477031 -> 1763584
Startcorner = 1
xOff = 349 yOff = 344 1:485814:1923081:487908 -> 2896804
Startcorner = 1
xOff = 350 yOff = -7 1:488958:508383:491058 -> 1488400
Startcorner = 1
xOff = 352 yOff = -336 1:494897:1891297:497009 -> 2883204
Startcorner = 1
xOff = 352 yOff = -156 1:494717:1030537:496829 -> 2022084
Startcorner = 1
xOff = 365 yOff = 305 1:531501:1797671:533691 -> 2862864
Startcorner = 1
xOff = 366 yOff = -21 1:534748:597571:536944 -> 1669264
Startcorner = 1
xOff = 367 yOff = -141 1:537797:1030507:539999 -> 2108304
Startcorner = 1
xOff = 368 yOff = -220 1:540813:1381065:543021 -> 2464900
Startcorner = 1
xOff = 371 yOff = -322 1:549774:1918869:552000 -> 3020644
Startcorner = 1
xOff = 371 yOff = -76 1:549528:797601:551754 -> 1898884
Startcorner = 1
xOff = 372 yOff = 258 1:552163:1589605:554395 -> 2696164
Startcorner = 1
xOff = 392 yOff = -316 1:613797:2002857:616149 -> 3232804
Startcorner = 1
xOff = 399 yOff = -237 1:635845:1615915:638239 -> 2890000
Startcorner = 1
xOff = 403 yOff = -52 1:648480:826385:650898 -> 2125764
Startcorner = 1
xOff = 404 yOff = -46 1:651699:808293:654123 -> 2114116
Startcorner = 1 xOff
= 412 yOff = -286 1:678027:1946597:680499 -> 3305124
Startcorner = 1
xOff = 420 yOff = 378 1:703963:2549653:706483 -> 3960100
Startcorner = 1
xOff = 421 yOff = 284 1:707418:1990353:709944 -> 3407716
Startcorner = 1
xOff = 431 yOff = -277 1:742029:2002779:744615 -> 3489424
Startcorner = 1
xOff = 432 yOff = 180 1:745021:1500265:747613 -> 2992900
Startcorner = 1
xOff = 432 yOff = 348 1:744853:2436025:747445 -> 3928324
Startcorner = 1
xOff = 433 yOff = 179 1:748479:1500267:751077 -> 2999824
Startcorner = 1
xOff = 434 yOff = -1 1:752124:755163:754728 -> 2262016
Startcorner = 1
xOff = 438 yOff = 105 1:765958:1181359:768586 -> 2715904
Startcorner = 1
xOff = 442 yOff = -255 1:780386:1940959:783038 -> 3504384
Startcorner = 1
xOff = 451 yOff = -372 1:812624:2706769:815330 -> 4334724
Startcorner = 1
xOff = 453 yOff = 24 1:819454:911977:822172 -> 2553604
Startcorner = 1
xOff = 453 yOff = 60 1:819418:1054609:822136 -> 2696164
Startcorner = 1
xOff = 456 yOff = 204 1:830173:1744633:832909 -> 3407716
Startcorner = 1
xOff = 457 yOff = 413 1:833613:3030255:836355 -> 4700224
Startcorner = 1
xOff = 464 yOff = -100 1:859893:1270329:862677 -> 2992900
Startcorner = 1
xOff = 464 yOff = 224 1:859569:1895681:862353 -> 3617604
Startcorner = 1
xOff = 467 yOff = -256 1:871212:2088537:874014 -> 3833764
Startcorner = 1
xOff = 467 yOff = -121 1:871077:1380867:873879 -> 3125824
Startcorner = 1
xOff = 473 yOff = 405 1:893093:3086239:895931 -> 4875264
Startcorner = 1
xOff = 475 yOff = -232 1:901308:1997033:904158 -> 3802500
Startcorner = 1
xOff = 476 yOff = 294 1:904583:2374093:907439 -> 4186116
Startcorner = 1
xOff = 477 yOff = 183 1:908503:1744675:911365 -> 3564544
Startcorner = 1
xOff = 504 yOff = -132 1:1014685:1615705:1017709 -> 3648100
(...)
Process finished with exit code 0
Merci et bravo, Scott !
É.
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