Magic square of square

Question

I am trying to solve magic square of square First let me explain my approach when will be the magic square form if we have
$$\begin{array}{|c|c|c|} \hline A² &B²&C² \\ \hline D²&E²&F²\\ \hline G²&H²&I²\\ \hline \end{array}$$ Then we have equations like- $$A²+B²+C²=x...(1)$$ $$D²+E²+F²=x...(2)$$ $$G²+H²+I²=x...(3)$$ $$A²+E²+I²=x...(4)$$ $$G²+E²+C²=x...(5)$$ $$A²+D²+G²=x...(6)$$ $$B²+E²+H²=x...(7)$$ $$C²+F²+I²=x...(8)$$ By solving all equation I found that-

1. $B²+H²=A²+I²=C²+G²=D²+F²=2E²$
   2.$H²+I²=C²+E²=A²+D²$
   3.$B²+C²=E²+I²=D²+G²$
   4.$G²+H²=A²+E²=C²+F²$
   5.$A²+B²=E²+G²=F²+I²$

From above equation we can find the value of $A²,C²,G²,I²$. Which are- $$2A²=H²+F²$$ $$2C²=D²+H²$$ $$2G²=B²+F²$$ $$2I²=B²+D²$$

So magic square can be made as $$\begin{array}{|c|c|c|} \hline (F²+H²)/2&B²&(D²+H²)/2\\ \hline D²&(D²+F²)/2 = (B²+H²)/2 &F²\\ \hline (F²+B²)/2&H²&(B²+D²)/2\\ \hline \end{array}$$
They are like Pythagorean triplets where $A,C,G,I,E$ are hypotenuse. So can pythagorean triplets like $((a,b,c) => (a²+b²=c²))$ :- $$(F,H,A), (B,D,I), (D,H,C), (B,F,G), (B,H,E), (D,F,E)$$ is exist- https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity
Example - if i have triplets like $(63,16,65)$ and $(33,56,65)$ then there should be other triplets like $(63,33,C_1),\ (16,33,C_2),\ (63,56,C_3) ,\ (16,56,C_4)$

Answer 1

Here are some "candidates" where $C \le 100,001.\quad$ They do not appear to be "complete" but there are no others under $100,000$ where $C$ is odd.

$$ (957,3724,3845) \quad (957,124,965) \quad (3843,124,3845) \\ \\ (2375,4200,4825 ) \quad (3367,4200,5383 ) \quad (3367,3456,4825 ) \\ (4697,1104,4825 ) \\ \\ (707,5076,5125) \quad (2813,4284,5125) \quad (3325,3900,5125) \\ (4675,4284,6341) \quad (4675,2100,5125) \\ \\ (4123,8364,9325 ) \quad (4123,4836,6355 ) \quad (6875,6300,9325 ) \\ (7973,4836,9325 ) \\ \\ (215,23112,23113) \quad (215,912,937) \quad (23095,912,23113) \\ \\ (13725,23452,27173) \quad (22755,23452,32677) \quad (22755,14852,27173) \\ \\ (8613,33516,34605) \quad (8613,1116,8685) \quad (34587,1116,34605) \\ \\ (6141,35380,35909) \quad (6141,23140,23941) \quad (27459,23140,35909) \\ \\ (32881,22920,40081) \quad (39919,22920,46031) \quad (39919,3600,40081) \\ \\ (27615,29072,40097) \quad (36735,29072,46847) \quad (36735,16072,40097) \\ \\ (21375,37800,43425) \quad (30303,37800,48447) \quad (30303,31104,43425) \\ (42273,9936,43425) \\ \\ (6363,45684,46125) \quad (25317,38556,46125) \quad (29925,35100,46125) \\ (42075,38556,57069) \quad (42075,18900,46125) \\ \\ (7015,46248,46777) \quad (7015,4752,8473) \quad (46535,4752,46777) \\ \\ (9951,51040,52001) \quad (40449,32680,52001) \quad (5871,47560,47921) \\ (32079,35600,47921) \\ \\ (38675,45732,59893) \quad (38675,27132,47243) \quad (53395,27132,59893) \\ \\ (5325,62900,63125) \quad (29325,55900,63125) \quad (29325,22724,37099) \\ (45453,43804,63125) \quad (58893,22724,63125) \quad (61875,12500,63125) \\ \\ (16779,62620,64829) \quad (16779,46060,49021) \quad (45621,46060,64829) \\ \\ (2737,76416,76465)\quad (2737,9384,9775) \quad (30257,70224,76465)\\ (72593,24024,76465) \quad (75887,9384,76465) \\ \\ (37107,75276,83925) \quad (37107,43524,57195) \quad (61875,56700,83925) \\ (71757,43524,83925) \\ \\ (23925,93100,96125) \quad (23925,3100,24125) \quad (29877,91364,96125) \\ (82677,49036,96125) \quad (96075,3100,96125) \quad $$

Answer 2

I'm sending you a program that generates candidate triples over a possible C-domain of $500$ at a time. It generates triples for a given C-value and also triples that can be formed from different elements of column a or b such as

$$ (31,480,481) \\ (480, 360, 600)\quad b+++\\ (360, 480, 600)\quad b+++\\ (319, 360, 481) \\ ***\\ (720, 638, 962 )\\ (720, 960, 1200)\quad a+++\\ (960, 720, 1200) \quad a+++\\ (960, 62, 962) $$