The smallest squares containingk 67's :
676 = 26²,   35676729 = 5973²,   36767679001 = 191749²,
16744676784676 = 4092026²,   26796767067346761 = 163697181².

67² is the third square which is the sum of 7 fifth powers : 1⁵ + 2⁵ + 2⁵ + 2⁵ + 3⁵ + 4⁵ + 5⁵.

67² = 4489, non-decreasing sequence of digits.

67² = 4489, 4 + 4 + 8 + 9 = 5²,
67² = 4489, 4³ + 4³ + 8³ + 9³ = 37²,
67² = 4489, 4² + 48² + 9² = 49².

67² + 68² + 69² + ... + 159² = 160² + 161² + 162² + ... + 198².

67^k + 209^k + 233^k + 275^k are squares for k = 1,2,3 (28², 422², 6548²).

Komachi equations:
67² = 1 * 2 / 3 * 4 * 5 * 6 * 7 * 8 + 9 = - 1 - 2 - 3 + 4567 - 8 * 9
  = - 1 * 2 * 3 + 4567 - 8 * 9 = 9 + 8 * 7 * 6 * 5 * 4 / 3 * 2 */ 1
  = - 9 * 8 + 76 * 5 * 4 * 3 + 2 - 1 = 98 + 76 - 5 + 432 * 10
  = - 9 * 8 + 7 * 654 + 3 - 2 * 10 = - 98 + 7 * 654 - 3 + 2 + 10
  = - 9 + 876 * 5 + 4 * 32 - 10,
67² = - 1² + 2² + 34² + 56² + 7² + 8² + 9² = 9² * 8² - 7² - 6² - 5² - 4² * 3² - 21².

(67² - 5) = (8² - 5)(9² - 5),
(67² - 7) = (5² - 7)(16² - 7) = (3² - 7)(4² - 7)(16² - 7),
(67² - 9) = (5² - 9)(17² - 9) = (7² - 9)(11² - 9) = (4² - 9)(5² - 9)(7² - 9).

274² = 44² + 45² + 46² + ... + 67²,
67² + 68² + 69² + ... + 116² = 655²,
67² + 68² + 69² + ... + 8516² = 453765².

(1 + 2 + ... + 11)(12)(13 + 14 + ... + 67) = 1320²,
(1 + 2 + ... + 18)(19 + 20 + ... + 37)(38 + 39 + ... + 67) = 11970²,
(1 + 2 + ... + 25)(26 + 27 + 28)(29 + 30 + ... + 67) = 7020²,
(1 + 2 + ... + 25)(26 + 27 + ... + 49)(50 + 51 + ... + 67) = 17550²,
(1 + 2 + ... + 25)(26 + 27 + ... + 52)(53 + 54 + ... + 67) = 17550²,
(1 + 2 + ... + 32)(33 + 34 + ... + 42)(43 + 44 + ... + 67) = 16500².

(1² + 2² + ... + 4²)(5² + 6² + ... + 12²)(13² + 13² + ... + 43²)(44² + 45² + ... + 67²) = 6115680²,
(1² + 2² + ... + 6²)(7² + 8² + ... + 29²)(30² + 31² + ... + 53²)(54² + 55² + ... + 67²) = 41039544²,
(1² + 2² + ... + 14²)(15² + 16² + ... + 21²)(22² + 23² + ... + 26²)(27² + 28² + ... + 67²) = 25468380²,
(1² + 2² + ... + 14²)(15² + 16² + ... + 29²)(30² + 31² + ... + 53²)(54² + 55² + ... + 67²) = 129363780²,
(1² + 2² + ... + 14²)(15² + 16² + ... + 62²)(63² + 64² + ... + 66²)(67²) = 78069740².

67² = 4489 appears in the decimal expressions of π and e:
  π = 3.14159•••4489••• (from the 4924th digit),
  e = 2.71828•••4489••• (from the 1188th digit).

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Page of Squares : First Upload February 9, 2004 ; Last Revised September 5, 2011
by Yoshio Mimura, Kobe, Japan