The smallest squares containing k 39's :
3969 = 632, 33953929 = 58272, 11323939396 = 1064142,
239393939841 = 4892792, 3939396393981321 = 627646112.
The 2nd integer which is the sum of 4 distinct squares: 12 + 22 + 32 + 52.
The 5th integer which is the sum of 4 squares in just 2 ways (see 38).
The 3rd integer which is the sum of 7 squares in just 3 ways (see 37).
1 / 39 = 0.0256..., 256 = 162.
The square root of 39 is 6.244..., 62 = 22 + 42 + 42.
392 = 1521, 1 + 5 + 2 + 1 = 32,
392 = 1521, 15 + 21 = 62.
392 = 13 + 23 + 83 + 103 = 13 + 43 + 53 + 113 = 43 + 63 + 83 + 93.
392 = 132 + 262 + 262, 622 + 622 + 312 = 932.
396 = 3518743761, 3 + 5 + 1 + 8 + 743 + 761 = 392,
397 = 137231006679, 13 + 723 + 100 + 6 + 679 = 1372 + 3 + 1 + 0066 + 79 = 392,
399 = 208728361158759,
2 + 08 + 72 + 8 + 3 + 611 + 58 + 759 = 2 + 08 + 72 + 836 + 1 + 1 + 587 + 5 + 9
= 2 + 08 + 728 + 3 + 6 + 1 + 1 + 5 + 8 + 759 = 2 + 0872 + 8 + 36 + 1 + 1 + 587 + 5 + 9
= 20 + 8 + 7 + 2 + 836 + 1 + 1 + 587 + 59 = 20 + 8 + 728 + 3 + 611 + 5 + 87 + 59
= 20 + 87 + 28 + 3 + 611 + 5 + 8 + 759 = 20 + 872 + 8 + 3 + 6 + 11 + 587 + 5 + 9
= 208 + 7 + 28 + 36 + 1158 + 75 + 9 = 208 + 7 + 28 + 361 + 158 + 759
= 208 + 72 + 8 + 3 + 6 + 1158 + 7 + 59 = 208 + 728 + 361 + 158 + 7 + 59 = 392.
392 = (12 + 22 + 32 + ... + 112) + (12 + 22 + 32 + ... + 142).
Komachi equations:
392 = 1 / 2 * 3456 * 7 / 8 + 9 = - 1 - 2 + 34 * 5 * 6 + 7 * 8 * 9,
392 = 9 - 8 + 76 * 5 * 4 - 3 + 2 + 1 = 987 - 6 + 543 - 2 - 1
= 9 + 8 * 765 / 4 + 3 - 21, and more 5 equations,
392 = 9 + 8 + 76 * 5 * 4 - 3 * 2 - 10 = - 9 + 8 + 7 * 6 * 54 / 3 * 2 + 10
= - 9 - 8 + 7 * 654 / 3 + 2 + 10,
392 = 12 + 22 * 342 - 562 + 72 + 82 - 92 = 122 * 32 + 42 + 562 / 72 + 82 + 92,
392 = 92 + 82 + 72 * 62 - 52 * 42 + 32 + 22 - 12,
392 = 13 * 23 + 33 * 43 - 53 - 63 + 73 + 83 - 93 = - 13 - 23 - 33 + 43 + 53 - 63 + 73 + 83 + 93,
392 = 93 + 83 + 73 - 63 + 53 + 43 - 33 - 23 - 13 = - 93 + 83 + 73 - 63 - 53 + 43 * 33 + 23 * 13
= - 93 + 83 + 73 - 63 - 53 + 43 * 33 + 23 / 13.
(392 - 3) = (32 - 3)(162 - 3) = (62 - 3)(72 - 3),
(392 + 9) = (12 + 9)(122 + 9) = (52 + 9)(62 + 9),
(392 - 9) = (42 - 9)(152 - 9).
1382 = 172 + 182 + 192 + ... + 392,
1432 = 72 + 82 + 92 + ... + 392.
(1 + 2 + ... + 12)(13 + 14 + ... + 39) = 2342,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 39) = 6302,
(1 + 2 + ... + 9)(10 + 11 + ... + 14)(15 + 16 + ... + 39) = 13502,
(1 + 2 + ... + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 39) = 16382,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 39) = 27722,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 39) = 37802,
(1 + 2 + ... + 27)(28 + 29 + ... + 35)(36 + 37 + ... + 39) = 37802,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 39) = 41582.
13 + 23 + ... + 393 = (1 + 2 + ... + 39)2 = 7802,
(13 + 23 + ... + 243)(253 + 263 + ... + 393) = 2160002.
392 = 1521 appears in the decimal expressions of π and e:
π = 3.14159•••1521••• (from the 1315th digit),
(1521 is the 6th 4-digit square in the expr. of π)
e = 2.71828•••1521••• (from the 1758th digit).
(13 + 23 + ... + 243)(253 + 263 + ... + 393) = 2160002.
Page of Squares : First Upload December 8, 2003 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan