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:
\(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | ||||||||||||
\(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | ||||||||||||
\(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | ||||||||||||
\(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(\) | \(\ds \) | ||||||||||||
\(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(\) | \(\ds \) |
Proof
Taking the $\dbinom 5 3 = 10$ subsets of $3$ integers at a time:
\(\text {(1)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(=\) | \(\ds 187 \, 949 \, 817 \, 467 \, 526 \, 392 \, 100\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 709 \, 479 \, 110^2\) |
\(\text {(2)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(=\) | \(\ds 256 \, 195 \, 895 \, 861 \, 438 \, 330 \, 625\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 16 \, 006 \, 120 \, 575^2\) |
\(\text {(3)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 589 \, 467 \, 277 \, 868 \, 756 \, 256 \, 225\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 24 \, 278 \, 947 \, 215^2\) |
\(\text {(4)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(=\) | \(\ds 330 \, 583 \, 710 \, 532 \, 830 \, 876 \, 900\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 18 \, 181 \, 961 \, 130^2\) |
\(\text {(5)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 663 \, 855 \, 092 \, 540 \, 148 \, 802 \, 500\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 25 \, 765 \, 385 \, 550^2\) |
\(\text {(6)}: \quad\) | \(\ds 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 732 \, 101 \, 170 \, 934 \, 060 \, 741 \, 025\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 27 \, 057 \, 368 \, 145^2\) |
\(\text {(7)}: \quad\) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(=\) | \(\ds 348 \, 256 \, 226 \, 963 \, 544 \, 806 \, 916\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 18 \, 661 \, 624 \, 446^2\) |
\(\text {(8)}: \quad\) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 681 \, 527 \, 608 \, 970 \, 862 \, 732 \, 516\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 106 \, 083 \, 754^2\) |
\(\text {(9)}: \quad\) | \(\ds 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 749 \, 773 \, 687 \, 364 \, 774 \, 671 \, 041\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 27 \, 381 \, 995 \, 679^2\) |
\(\text {(10)}: \quad\) | \(\ds 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747\) | \(\) | \(\ds \) | |||||||||||
\(\, \ds + \, \) | \(\ds 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347\) | \(=\) | \(\ds 824 \, 161 \, 502 \, 036 \, 167 \, 217 \, 316\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 28 \, 708 \, 213 \, 146^2\) |
$\blacksquare$
Sources
- Oct. 1995: Stan Wagon: Quintuples with Square Triplets (Math. Comp. Vol. 64, no. 212: pp. 1755 – 1756) www.jstor.org/stable/2153384
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $26,072,323,311,568,661,931$