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72

The smallest squares containingk 72's :
729 = 272,   7230721 = 26892,   2727241729 = 522232,
1723472721721 = 13128112,   9972107277272721 = 998604392.

722 = 5184, a zigzag squares with different digits.

72 is the sum of m squares for m = 2, 3, ..., 58.

722 = 4! + 5! + 7!

722± 5 are primes.

722 = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + ... + 21)
  = (1 + 2)(3)(4)(5 + 6 + 7)(8) = (1 + 2)(3 + 4 + 5)(6)(7 + 8 + 9).

722 = 123 + 123 + 123 = 64 + 64 + 64 + 64 = 25 + 25 + 45 + 45 + 45 + 45 + 45.

22 + 92 + 162 + 232 + ... + 722 = 143722 .

728 = 722204136308736, 7 + 2 + 2 + 2041 + 3 + 6 + 3087 + 36 = 722,
    7 + 2 + 2 + 2041 + 36 + 3087 + 3 + 6 = 722 + 2 + 04136 + 308 + 7 + 3 + 6 = 722,
729 = 51998697814228992, 5 + 1998 + 6 + 9 + 781 + 4 + 2289 + 92 = 722,
    51+9+9+8+697+81+4228+9+92 = 51+9+9+8+697+81+4228+99+2
  = 51+9+98+697+81+4228+9+9+2 = 51+99+8+697+81+4228+9+9+2 = 722.

Komachi equations:
722 = 12 * 3 / 4 * 56 / 7 * 8 * 9 = 9 * 8 * 7 * 6 + 5 * 432 */ 1
  = 98 + 765 + 4321,
722 = 12 + 22 - 32 + 42 - 52 - 62 + 72 + 82 * 92 = 122 * 32 / 42 - 52 * 62 + 782 - 92
  = - 12 - 22 + 32 - 42 + 52 + 62 - 72 + 82 * 92 = - 12 + 22 * 32 * 42 - 52 + 672 + 82 + 92
  = - 122 * 32 / 42 - 52 * 62 + 782 + 92 = 92 * 82 + 72 - 62 - 52 + 42 - 32 + 22 + 12
  = 92 * 82 - 72 + 62 + 52 - 42 + 32 - 22 - 12 = 92 * 82 + 72 - 62 / 542 * 32 * 212
  = 92 * 82 + 72 / 62 * 542 - 32 * 212 = 92 * 82 + 72 / 62 * 542 / 32 - 212
  = 92 * 82 - 72 + 62 / 542 * 32 * 212 = 92 * 82 - 72 / 62 * 542 + 32 * 212
  = 92 * 82 - 72 / 62 * 542 / 32 + 212,
722 = 93 - 83 + 73 - 63 + 543 / 33 + 23 - 103,
722 = - 14 * 24 - 34 - 44 - 54 + 64 + 74 - 84 + 94 = 94 - 84 + 74 + 64 - 54 - 44 - 34 - 24 */ 14.

Cubic polynomial (X + 352)(X + 722)(X + 962) = X3 + 1252X2 + 80882X + 2419202,
Cubic polynomial (X + 652)(X + 722)(X + 47042) = X3 + 47052X2 + 4563122X + 220147202.

(722 + 3) = (22 + 3)(42 + 3)(62 + 3).

(1 + 2)(3)(4)(5 + 6 + 7)(8) = (1 + 2)(3 + 4 + 5)(6)(7 + 8 + 9) = 722.

(1 + 2)(3 + 4 + ... + 47)(48 + 49 + ... + 72) = 22502,
(1 + 2 + ... + 4)(5 + 6 + ... + 55)(56 + 57 + ... + 72) = 40802,
(1 + 2 + ... + 9)(10 + 11 + ... + 14)(15 + 16 + ... + 72) = 26102,
(1 + 2 + ... + 9)(10 + 11 + ... + 47)(48 + 49 + ... + 72) = 85502,
(1 + 2 + ... + 20)(21 + 22 + ... + 44)(45 + 46 + ... + 72) = 163802,
(1 + 2 + ... + 39)(40 + 41 + ... + 44)(45 + 46 + ... + 72) = 163802,
(1 + 2 + ... + 39)(40 + 41 + ... + 59)(60 + 61 + ... + 72) = 257402,
(1 + 2 + ... + 44)(45 + 46 + ... + 59)(60 + 61 + ... + 72) = 257402.

(12)(22 + 32 + ... + 102)(112 + 122 + ... + 532)(542 + 552 + ... + 722) = 12156962,
(12)(22 + 32 + ... + 102)(112 + 122 + ... + 672)(682 + 692 + ... + 722) = 9804002,
(12 + 22 + 32)(42)(52 + 62 + ... + 112)(122 + 132 + ... + 722) = 1161442,
(12 + 22 + ... + 52)(62)(72 + 82 + ... + 392)(402 + 412 + ... + 722) = 20763602,
(12 + 22 + ... + 52)(62 + 72 + ... + 102)(112 + 122 + ... + 532)(542 + 552 + ... + 722) = 83579102,
(12 + 22... + 52)(62 + 72 + ... + 102)(112 + 122 + ... + 672)(682 + 692 + ... + 722) = 67402502,
(12 + 22 + ... + 112)(122 + 132 + ... + 162)(172)(182 + 192 + ... + 722) = 42579902.

722 = 5184 appears in the decimal expressions of π and e:
  π = 3.14159•••5184••• (from the 6434th digit),
  e = 2.71828•••5184••• (from the 4551st digit).


Page of Squares : First Upload February 16, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan