5 Numbers such that Sum of any 3 is Square

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Theorem

This set of $5$ integers has the property that the sum of any $3$ of them is square:

\(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(\) \(\displaystyle \)
\(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(\) \(\displaystyle \)


Proof

Taking the $\dbinom 5 3 = 10$ subsets of $3$ integers at a time:


\((1):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(=\) \(\displaystyle 187 \, 949 \, 817 \, 467 \, 526 \, 392 \, 100\)
\(\displaystyle \) \(=\) \(\displaystyle 13 \, 709 \, 479 \, 110^2\)


\((2):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(=\) \(\displaystyle 256 \, 195 \, 895 \, 861 \, 438 \, 330 \, 625\)
\(\displaystyle \) \(=\) \(\displaystyle 16 \, 006 \, 120 \, 575^2\)


\((3):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 589 \, 467 \, 277 \, 868 \, 756 \, 256 \, 225\)
\(\displaystyle \) \(=\) \(\displaystyle 24 \, 278 \, 947 \, 215^2\)


\((4):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(=\) \(\displaystyle 330 \, 583 \, 710 \, 532 \, 830 \, 876 \, 900\)
\(\displaystyle \) \(=\) \(\displaystyle 18 \, 181 \, 961 \, 130^2\)


\((5):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 663 \, 855 \, 092 \, 540 \, 148 \, 802 \, 500\)
\(\displaystyle \) \(=\) \(\displaystyle 25 \, 765 \, 385 \, 550^2\)


\((6):\quad\) \(\displaystyle 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 732 \, 101 \, 170 \, 934 \, 060 \, 741 \, 025\)
\(\displaystyle \) \(=\) \(\displaystyle 27 \, 057 \, 368 \, 145^2\)


\((7):\quad\) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(=\) \(\displaystyle 348 \, 256 \, 226 \, 963 \, 544 \, 806 \, 916\)
\(\displaystyle \) \(=\) \(\displaystyle 18 \, 661 \, 624 \, 446^2\)


\((8):\quad\) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 681 \, 527 \, 608 \, 970 \, 862 \, 732 \, 516\)
\(\displaystyle \) \(=\) \(\displaystyle 26 \, 106 \, 083 \, 754^2\)


\((9):\quad\) \(\displaystyle 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 749 \, 773 \, 687 \, 364 \, 774 \, 671 \, 041\)
\(\displaystyle \) \(=\) \(\displaystyle 27 \, 381 \, 995 \, 679^2\)


\((10):\quad\) \(\displaystyle 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) \(\) \(\displaystyle \)
\(\, \displaystyle + \, \) \(\displaystyle 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) \(=\) \(\displaystyle 824 \, 161 \, 502 \, 036 \, 167 \, 217 \, 316\)
\(\displaystyle \) \(=\) \(\displaystyle 28 \, 708 \, 213 \, 146^2\)

$\blacksquare$


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