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:

+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 3| 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+.
| 25 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | 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:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |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| 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 25 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 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 | | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
2 | | 9 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118|   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 12| 1825| 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 | | 1015| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
2 | | 9 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 8 | 13| 19| 26| 34| 43| 53| 64| 76| 89|103|118|   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 712| 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 | 1015| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
2 | | 9 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | 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| 243241| 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 | 1015| 21| 28| 36| 45| 55| 66| 78| 91|105|120|   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
2 | | 9 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | 8 | 13| 19| 26| 34| 4353| 64| 76| 89|103|118|   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
7 | 12| 18| 25| 33| 42| 5263| 75| 88|102|117|   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
11| 17| 24| 32| 41| 51| 62| 74| 87|101|116|   |   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 16| 23| 31| 40| 50| 6173| 86|100|115|   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
223039| 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| 737475767778798081| 82|123|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |153|108| 71| 42| 2122232425| 26| 51| 84|125|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |151|106| 69| 40| 196 | 1 | 2 | 11| 28| 53| 86|127|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |150|105| 68| 39| 185 | 4 | 3 | 12| 29| 54| 87|128|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |149|104| 67| 38| 1716151413| 30| 55| 88|129|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |147|102| 656463626160595857| 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| 737475767778798081| 82|123|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |153|108| 71| 42| 2122232425| 26| 51| 84|125|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |152|107| 70| 41| 20| 7 | | 9 | 10| 27| 52| 85|126|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |151|106| 69| 40| 191 | 2 | 11| 28| 53| 86|127|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |149|104| 67| 38| 1716151413| 30| 55| 88|129|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |147|102| 656463626160595857| 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| 737475767778798081| 82|123|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |153|108| 71| 42| 21| 22232425| 26| 51| 84|125|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 2752| 85|126|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |151|106| 69| 40| 191 | 2 | 11| 28| 53| 86|127|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |149|104| 67| 38| 1716151413| 30| 55| 88|129|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |147|102| 656463626160595857| 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| 737475767778798081| 82|123|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |153|108| 71| 42| 21| 22232425| 26| 51| 84|125|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |152|107| 70| 41| 20| 7 | 8 | 9 | 102752| 85|126|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |151|106| 69| 40| 191 | 2 | 11| 28| 53| 86|127|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |149|104| 67| 38| 1716151413| 30| 55| 88|129|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |147|102| 656463626160595857| 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| 737475767778798081| 82|123|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |153|108| 71| 42| 21| 22232425| 26| 51| 84|125|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |152|107| 70| 4120| 7 | 8 | 9 | 102752| 85|126|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |151|106| 69| 40| 191 | 2 | 11| 28| 5386|127|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |149|104| 67| 38| 1716151413| 30| 55| 88|129|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130|   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |147|102| 656463626160595857| 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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