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93

The smallest squares containingk 93's :
1936 = 442,   81793936 = 90442,   932935936 = 305442,
2939359375936 = 17144562,   1939349319345936 = 440380442.

932 = 8649, a square with different digits.

932 = 8649, 83 + 63 + 43 + 93 = 392,
932 = 8649, 8 + 64 + 9 = 92.

932 = 312 + 622 + 622, 262 + 262 + 132 = 392.

932 = (12 + 23 + 34) +(42 + 53 + 64) + (72 + 83 + 94) = 932.

10k + 44k + 57k + 58k are squares for k = 1,2,3 (132, 932, 6832).
22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).

Komachi equations:
932 = 9 + 8 / 7 * 6 * 5 * 4 * 3 * 21 = 9 + 8 * 765 + 4 * 3 * 210
= - 9 + 8 - 76 * 5 + 43 * 210,
932 = 982 - 72 * 62 + 52 + 42 / 32 * 212 = 982 - 72 * 62 + 52 * 42 + 32 + 22 * 102.

(932 + 3) = (92 + 3)(102 + 3) = (22 + 3)(32 + 3)(102 + 3),
(932 - 3) = (32 - 3)(382 - 3),   (932 - 9) = (72 - 9)(152 - 9).

4952 = 442 + 452 + 462 + ... + 932.

A 3-by-3 magic square consisting of different squares with constant 932:

82532762
642522432
672562322

932 = 8649 appears in the decimal expressions of π and e:
  π = 3.14159•••8649••• (from the 2990th digit),
  e = 2.71828•••8649••• (from the 16829th digit).

(1 + 2)(3 + 4 + 5)(6 + 7 + ... + 93) = 3962,
(1 + 2)(3 + 4 + ... + 42)(43 + 44 + ... + 93) = 30602,
(1 + 2 + 3 + 4)(5 + 6 + ... + 16)(17 + 18 + ... + 93) = 23102,
(1 + 2 + 3 + 4)(5 + 6 + ... + 25)(26 + 27 + ... + 93) = 35702,
(1 + 2 + 3 + 4)(5 + 6 + ... + 84)(85 + 86 + ... + 93) = 53402,
(1 + 2 + ... + 6)(7 + 8 + ... + 51)(52 + 53 + ... + 93) = 91352,
(1 + 2 + ... + 8)(9 + 10 + ... + 42)(43 + 44 + ... + 93) = 104042,
(1 + 2 + ... + 9)(10 + 11 + ... + 25)(26 + 27 + ... + 93) = 71402,
(1 + 2 + ... + 13)(14 + 15 + ... + 25)(26 + 27 + ... + 93) = 92822,
(1 + 2 + ... + 13)(14 + 15 + ... + 49)(50 + 51 + ... + 93) = 180182,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + ... + 93) = 188102,
(1 + 2 + ... + 35)(36 + 37 + ... + 51)(52 + 53 + ... + 93) = 365402,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 93) = 508202,
(1 + 2 + ... + 76)(77)(78 + 79 + ... + 93) = 175562,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 93) = 162362.

(12 + 22 + 32 + 42)(52 + 62 + ... + 122)(132 + 142 + ... + 432)(442 + 452 + ... + 932) = 110484002.


Page of Squares : First Upload March 15, 2004 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan