There are two 12s in the first diagram. Should one of them be 13? On Mon, Jun 29, 2020 at 5:55 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, today we'll look for « prime squares » and 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 | 12| 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 snail-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| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ (...snip -- see the rest here in full colors: https://bit.ly/2NGeAvU ) (1) Can you illustrate all geometric side-lengths of the "prime squares"? (2) Can you find all "square squares" constructible on a B grid? Best, É.
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