The smallest squares containingk 90's :
900 = 302, 5904900 = 24302, 908901904 = 301482,
6909049049049 = 26285072, 690904904904900 = 262850702.
(61 / 90)2 = 0.459382716... (Komachic).
90 is the sum of m squares for m = 2, 3, ..., 76.
90 = (12)(22 + 32 + 42 + 52 + 62).
The first integer which is the sum of 4 squares in just 6 ways:
[1,2,2,9], [1,2,6,7], [1,3,4,8], [2,5,5,6], [3,3,6,6], [3,4,4,7].
The first integer which is the sum of 4 distinct squares in just 2 ways.
902 is the third square which is the sum of 5 cubes in just 4 ways.
902 = 4 x 5 + 5 x 6 + 6 x 7 + ... + 28 x 29.
902 = 4 x 5 x 6 + 5 x 6 x 7 + 6 x 7 x 8 + ... + 12 x 13 x 14.
902 = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 19) = (1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + ... + 12)
= (1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11) = (1 + 2)(3 + 4 + 5 + 6 + ... + 17)(18)
= (1 + 2)(3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11 + ... + 18) = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 16)
= (1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + ... + 40).
902 = (12 + 9)(32 + 9)(62 + 9) = (32 + 9)(212 + 9) = (33 + 9)(63 + 9).
902 = 83 + 93 + 193.
9k + 10k + 60k + 90k are squares for k = 1,2,3 (132, 1092, 9732).
498k + 942k + 3138k + 3522k are squares for k = 1,2,3 (902, 48362, 2748602).
762k + 1938k + 2334k + 3066k are squares for k = 1,2,3 (902, 43802, 2219402).
1010k + 1690k + 2570k + 2830k are squares for k = 1,2,3 (902, 43002, 2133002).
Komachi equations:
902 = 9 * 876 + 5 * 43 + 2 - 1 = 9 * 87 * 6 + 54 * 3 * 21
= 9 * 876 + 5 * 432 / 10 = 9 * 876 + 5 + 4 - 3 + 210,
902 = 12 / 2342 * 52 * 62 * 782 * 92 = 12 * 22 * 32 + 452 + 62 + 782 - 92
= 12 - 22 + 32 * 42 + 52 - 62 + 72 + 892 = 122 - 32 - 42 - 52 + 62 + 72 + 892
= 12 / 22 * 342 - 52 - 62 - 72 + 892 = - 122 + 32 + 42 * 52 * 62 - 782 - 92
= 92 + 872 + 62 + 52 * 42 + 32 + 22 + 12 = 92 * 82 + 72 - 62 + 542 - 32 - 22 */ 12
= 92 * 82 - 72 + 62 + 542 + 32 + 22 */ 12 = - 982 / 72 + 62 * 52 + 432 * 22 */ 12
= - 92 - 82 + 72 + 62 * 52 + 432 * 22 - 102,
902 = 93 + 83 + 763 / 53 * 43 / 323 * 103.
(902 - 6) = (32 - 6)(522 - 6), (902 - 8) = (52 - 8)(222 - 8).
12 + 22 + 32 + 42 + ... + 902 = 247065, which consists of different digits.
(1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + 9)(10 + 11 + ... + 90) = 13502,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 30)(31 + 32 + ... + 90) = 49502.
(12 + 22 + 32)(42 + ... + 632)(642 + ... + 722)(732 + ... + 902) = 773089802.
902 = 8100 appears in the decimal expressions of π and e:
π = 3.14159•••8100••• (from the 2879th digit),
e = 2.71828•••8100••• (from the 9804th digit).
Page of Squares : First Upload March 15, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan