Nested Magic Squares With Perfect Square Sums, Pythagorean Triples, and Borders Differences

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. Instead, considering entries as consecutive numbers, we considered consecutive odd numbers entries. This gives us perfect square sum magic squares. We worked with magic squares of orders 3 to 25. Finally, the result can be written in a symmetric way, i.e.,

T_{m x k}:= m^2 x k^2,

where m is the order of nested magic square, k is the order of sub-magic squares in each case, and T is the total entries sum. Also the entries sums are connected with Pythagorean triples and borders. In each case, the difference among consecutive borders sums is always a fixed value. Below is link for download the work and some examples:

nms-1nms-2nms-3nms-4nms-5nms-6nms-7nms-8nms-9

Link for download the work:

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