The smallest squares containingk 88's :
6889 = 83²,   2588881 = 1609²,   38888628804 = 197202²,
4888888032889 = 2211083²,   788088668888889 = 28072917².

88² = 7744.

88² = 7744, 77 + 44 = 11².

88⁸ = 3596345248055296, 3596 + 3452 + 48 + 0552 + 96 = 88².

88² = (1² + 7)(2² + 7)(9² + 7) = (1² + 7)(31² + 7).

88^k + 638^k + 1122^k + 1177^k are squares for k = 1,2,3 (55², 1749², 57475²).

Komachi equations: 88² = 9 * 876 - 5 * 4 / 3 * 21 = 9 * 8 + 7654 - 3 + 21,
88² = 1² * 23² * 4² - 5² + 67² - 8² * 9² = 1² + 2² - 3² * 4² - 5² + 6² - 7² + 89²
  = 98² + 7² - 6² * 5² + 4² - 32² - 1² = 98² - 7² * 6² + 5² - 4² - 3² + 2² - 10²
  = 98² - 7² * 6² - 5² + 4² + 3² + 2² - 10² = - 9² + 87² + 6² + 54² / 3² - 2² - 10²,
88² = - 9³ + 8³ - 7³ + 6³ + 5³ - 4³ + 3³ + 2³ * 10³,
88² = - 1⁴ * 2⁴ - 3⁴ + 4⁴ + 5⁴ - 6⁴ - 7⁴ + 8⁴ + 9⁴ = 9⁴ + 8⁴ - 7⁴ - 6⁴ + 5⁴ + 4⁴ - 3⁴ - 2⁴ */ 1⁴.

(88² - 2) = (9² - 2)(10² - 2) = (3² - 2)(4² - 2)(9² - 2),
(88² + 5) = (2² + 5)(4² + 5)(6² + 5).

88² = (1² + 2² + 3² + 4²) + (1² + 2² + 3² + ... + 28²).

(1)(2 + 3 + ... + 7)(8 + 9 + ... + 88) = 324²,
(1 + 2)(3 + 4 + ... + 7)(8 + 9 + ... + 88) = 540²,
(1 + 2 + ... + 6)(7)(8 + 9 + ... + 88) = 756²,
(1 + 2 + ... + 6)(7 + 8 + ... + 25)(26 + 27 + ... + 88) = 4788²,
(1 + 2 + ... + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 88) = 6006²,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 88) = 10164²,
(1 + 2 + ... + 12)(13 + 14 + ... + 39)(40 + 41 + ... + 88) = 13104²,
(1 + 2 + ... + 16)(17 + 18 + ... + 64)(65 + 66 + ... + 88) = 22032²,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 88) = 15246²,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 88) = 13860²,
(1 + 2 + ... + 44)(45 + 46 + ... + 76)(77 + 78 + ... + 88) = 43560².

88² = 7744 appears in the decimal expressions of π and e:
  π = 3.14159•••7744••• (from the 4965th digit),
  e = 2.71828•••7744••• (from the 291st digit).
  (7744 is the third 4-digit square in the expression of e.)

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Page of Squares : First Upload March 8, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan