Theorem: 5 Numbers such that Sum of any 3 is Square

This set of $5$ integers has the property that the sum of any $3$ of them is square:
{{begin-eqn}}
{{eqn | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | o = 
}}
{{eqn | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | o = 
}}
{{end-eqn}}

Proof:
Taking the $\dbinom 5 3 = 10$ subsets of $3$ integers at a time:
{{begin-eqn}}
{{eqn | n = 1
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | r = 187 \, 949 \, 817 \, 467 \, 526 \, 392 \, 100
}}
{{eqn | r = 13 \, 709 \, 479 \, 110^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 2
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | r = 256 \, 195 \, 895 \, 861 \, 438 \, 330 \, 625
}}
{{eqn | r = 16 \, 006 \, 120 \, 575^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 3
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 589 \, 467 \, 277 \, 868 \, 756 \, 256 \, 225
}}
{{eqn | r = 24 \, 278 \, 947 \, 215^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 4
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | r = 330 \, 583 \, 710 \, 532 \, 830 \, 876 \, 900
}}
{{eqn | r = 18 \, 181 \, 961 \, 130^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 5
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 663 \, 855 \, 092 \, 540 \, 148 \, 802 \, 500
}}
{{eqn | r = 25 \, 765 \, 385 \, 550^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 6
      | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 732 \, 101 \, 170 \, 934 \, 060 \, 741 \, 025
}}
{{eqn | r = 27 \, 057 \, 368 \, 145^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 7
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | r = 348 \, 256 \, 226 \, 963 \, 544 \, 806 \, 916
}}
{{eqn | r = 18 \, 661 \, 624 \, 446^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 8
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 681 \, 527 \, 608 \, 970 \, 862 \, 732 \, 516
}}
{{eqn | r = 26 \, 106 \, 083 \, 754^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 9
      | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 749 \, 773 \, 687 \, 364 \, 774 \, 671 \, 041
}}
{{eqn | r = 27 \, 381 \, 995 \, 679^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 10
      | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
      | o = 
}}
{{eqn | lo= +
      | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747
      | o = 
}}
{{eqn | lo= +
      | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347
      | r = 824 \, 161 \, 502 \, 036 \, 167 \, 217 \, 316
}}
{{eqn | r = 28 \, 708 \, 213 \, 146^2
}}
{{end-eqn}}
{{qed}}