The smallest squares containingk 73's :
7396 = 86²,   732736 = 856²,   16737373129 = 129373²,
7373733873444 = 2715462²,   4273737387373584 = 65373828².

73 is the sum of m squares for m = 2, 3, 4, ..., 59.

73 is the 4th prime for which Legendre Symbol (a/73) = 1 for a = 1, 2, 3, 4.

1 / 73 = 0.01369..., 1369 = 37².

73² = 5329 with different digits.

73² = 1! + 4! + 4! + 5! + 5! + 7!.

73² = 5329, 53 + 2 + 9 = 8².

73⁶ = 151334226289, 1 + 5133 + 42 + 2 + 62 + 89 = 73²,
73⁷ = 11047398519097,
    1 + 10 + 4 + 7 + 3 + 9 + 8 + 5190 + 97 =1 + 10 + 4 + 7 + 3 + 98 + 5190 + 9 + 7 = 73²,
    11 + 04 + 7 + 3 + 9 + 8 + 5190 + 97 = 11 + 04 + 7 + 3 + 98 + 5190 + 9 + 7 = 73².

22^k + 73^k + 130^k + 136^k are squares for k = 1,2,3 (19², 203², 2261²).
62^k + 73^k + 266^k + 440^k are squares for k = 1,2,3 (29², 523², 10229²).

Komachi Fraction : (65/73)² = 38025/47961.

Komachi equations:
73² = 9 + 8 * 76 * 5 / 4 / 3 * 21 = - 9 - 87 - 6 + 5432 - 1
  = - 9 + 87 * 65 + 4 - 321 = 9 * 8 * 76 - 5 - 4 * 32 - 10
  = 9 * 87 * 6 + 5 - 4 + 3 * 210 = - 9 - 8 - 76 + 5432 - 10,
73² = 1² + 2² * 3² / 4² * 56² / 7² + 8² * 9² = 12² * 3² / 4² + 56² / 7² + 8² * 9²
  = 12² * 3² / 4² * 56² / 7² + 8² + 9² = - 1² * 2² + 3² * 4² * 5² / 6² + 7² + 8² * 9²
  = - 1² + 2² * 34² + 5² * 6² - 7² - 8² - 9² = - 1² * 2² + 34² + 56² / 7² * 8² + 9²
  = 9² * 8² + 7² + 6² + 5² + 4² * 3² / 2² - 1² = 9² + 8² - 7² + 65² - 4² + 32² */ 1²
  = 9² - 8² + 76² - 5² * 4² + 3² * 2² - 10² = 98² / 7² + 65² - 4² + 32² - 10²,
73² = 9⁴ - 8⁴ - 7⁴ - 6⁴ + 54⁴ / 3⁴ / 2⁴ */ 1⁴.

(73² + 1) = (8² + 1)(9² + 1),   (73² + 7) = (4² + 7)(15² + 7).

357² = 25² + 26² + 27² + ... + 73².

73² + 74² + 75² + ... + 194² = 1525²,
73² + 74² + 75² + ... + 22873² = 1997277².

(1)(2 + 3 + ... + 10)(11 + 12 + ... + 73) = 378²,
(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + ... + 73) = 594²,
(1 + 2 + 3)(4 + 5 + ... + 10)(11 + 12 + ... + 73) = 882²,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 73) = 2058²,
(1 + 2 + 3)(4 + 5 + ... + 38)(39 + 40 + ... + 73) = 2940²,
(1 + 2 + 3)(4 + 6 + ... + 51)(52 + 53 + ... + 73) = 3300²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 34)(35 + 36 + ... + 73) = 3510²,
(1 + 2 + ... + 5)(6 + 7 + ... + 10)(11 + 12 + ... + 73) = 1260²,
(1 + 2 + ... + 5)(6 + 7 + ... + 38)(39 + 40 + ... + 73) = 4620²,
(1 + 2 + ... + 8)(9 + 10 + ... + 17)(18 + 19 + ... + 73) = 3276²,
(1 + 2 + ... + 10)(11 + 12 + ... + 25)(26 + 27 + ... + 73) = 5940²,
(1 + 2 + ... + 13)(14 + 15 + ... + 34)(35 + 36 + ... + 73) = 9828²,
(1 + 2 + ... + 16)(17 + 18 + ... + 28)(29 + 30 + ... + 73) = 9180²,
(1 + 2 + ... + 17)(18 + 19 + ... + 22)(23 + 24 + ... + 73) = 6120²,
(1 + 2 + ... + 17)(18 + 19 + ... + 34)(35 + 36 + ... + 73) = 11934²,
(1 + 2 + ... + 17)(18 + 19 + ... + 45)(46 + 47 + ... + 73) = 14994²,
(1 + 2 + ... + 21)(22 + 23 + ... + 69)(70 + 71 + ... + 73) = 12012²,
(1 + 2 + ... + 26)(27 + 28 + ... + 38)(39 + 40 + ... + 73) = 16380²,
(1 + 2 + ... + 39)(40 + 41 + ... + 56)(57 + 58 + ... + 73) = 26520²,
(1 + 2 + ... + 40)(41 + 42 + ... + 49)(50 + 51 + ... + 73) = 22140²,
(1 + 2 + ... + 43)(44 + 45 + ... + 55)(56 + 57 + ... + 73) = 25542²,
(1 + 2 + ... + 51)(52 + 53 + ... + 56)(57 + 58 + ... + 73) = 19890²,
(1 + 2 + ... + 65)(66 + 67 + ... + 69)(70 + 71 + ... + 73) = 12870²,
(1 + 2 + ... + 72)(73) = 438².

(1² + 2² + ... + 24²)(25² + 26² + ... + 73²) = 24990²,
(1² + 2² + ... + 10²)(11² + 12² + ... + 15²)(16² + 17² + ... + 48²)(49² + 50² + ... + 73²) = 33795300²,
(1² + 2² + ... + 34²)(35² + 36² + ... + 42²)(43² + 44² + ... + 50²)(51² + 52² + ... + 73²) = 502513200².

73² = 5329 appears in the decimal expressions of π and e:
  π = 3.14159•••5329••• (from the 3331st digit),
  e = 2.71828•••5329••• (from the 22553rd digit).

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Page of Squares : First Upload February 16, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan