The smallest squares containing k 28's :
289 = 17²,   2862864 = 1692²,   2282832841 = 47779²,
282728285284 = 531722²,   28285282828881 = 5318391².

The first integer which is the sum of 4 squares in just 3 ways:
  1² + 1² + 1² + 5² = 1² + 3² + 3² + 3² = 2² + 2² + 2² + 4²

The 3rd integer which is the sum of 7 squares in just two ways.
The 1st integer which is the sum of 13 squares in just two ways (see 20).

28 = (1² + 2² + 3² + ... + 7²) / (1² + 2²).

28² = 784 is a zigzag square.

28² is the first square which is the sum of 7 cubes.

28² = (1² + 3)(2² + 3)(5² + 3).

28² = 1³ + 3³ + 3³ + 9³ = 2⁴ + 4⁴ + 4⁴ + 4⁴.

28⁹ = 10578455953408, 1 + 057 + 84 + 5 + 595 + 34 + 08 = 28² and more 8 equations,
28¹⁰ = 296196766695424, 2 + 9 + 6 + 1 + 9 + 67 + 6 + 669 + 5 + 4 + 2 + 4 = 28² and more 28 equations,
28¹¹ = 8293509467471872, 8 + 2 + 9 + 3 + 5 + 09 + 4 + 6 + 7 + 4 + 718 + 7 + 2 = 28² and more 29 equations.

Cubic Polynomial (X + 9²)(X + 28²)(X + 432²) = X³ + 433²X² + 12708²X + 108864²

(28² - 2) = (5² - 2)(6² - 2),   (28² + 8) = (4² + 8)(5² + 8).

77² = 18² + 19² + 20² + ... + 28².

28² + 29² + 30² + ... + 123² = 788²,
28² + 29² + 30² + ... + 77² = 385².

28² = (1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11),
28² = (1 + 2 + 3 + 4 + 5 + 6 + 7)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³.

17^k + 91^k + 299^k + 377^k are squares for k = 1,2,3 (28², 490², 9004²).
67^k + 209^k + 233^k + 275^k are squares for k = 1,2,3 (28², 422², 6548²).

Komachi Fraction : (28/197)² = 7056/349281.

Komachi equations:
28² = 12 * 3 * 4 * 56 * 7 / 8 / 9 = 1 * 234 + 5 + 67 * 8 + 9
  = 1 * 234 + 567 - 8 - 9, and more 24 equations,
28² = 9 + 8 * 7 + 6 * 5 * 4 * 3 * 2 - 1 = 9 * 8 - 7 + 6 * 5 * 4 * 3 * 2 - 1
  = 9 * 87 + 6 + 5 - 4 - 3 - 2 - 1, and more 30 equations,
28² = 9 * 87 + 6 - 5 + 4 + 3 * 2 - 10 = 9 * 87 + 6 - 5 + 4 * 3 - 2 - 10
  = 9 * 87 + 6 - 5 - 4 - 3 * 2 + 10, and more 26 equations,
28² = 1² + 2² - 3² - 4² + 5² * 6² + 7² - 8² - 9²,
28² = 9² * 8² * 7² / 6² * 5² / 4² / 3² - 21² = 98² / 7² * 6² * 5² / 4² / 3² - 21²
  = 98² / 7² + 6² - 5² + 4² * 3² * 2² + 1² = 98² / 7² - 6² - 5² * 4² + 32² * 1²
  = 98² / 7² - 6² - 5² * 4² + 32² / 1²,
28² = 9⁴ + 8⁴ + 7⁴ - 6⁴ - 5⁴ - 4⁴ - 3⁴ - 2⁴ - 10⁴.

(1)(2 + 3 + ... + 25)(26 + 27 + 28) = 162²,
(1 + 2)(3)(4 + 5 + ... + 28) = 60²,
(1 + 2)(3 + 4 + ... + 10)(11 + 12 + ... + 28) = 234²,
(1 + 2 + ... + 5)(6 + 7 + ... + 11)( 12 + 13 + ... + 28) = 510²,
(1 + 2 + ... + 6)(7 + 8 + ... + 20)( 21 + 22 + ... + 28) = 882²,
(1 + 2 + ... + 8)(9)(10 + 11 + ... + 28) = 342²,
(1 + 2 + ... + 8)(9 + 10 + ... + 25)(26 + 27 + 28) = 918²,
(1 + 2 + ... + 14)(15 + 16 + ... + 20)(21 + 22 + ... + 28) = 1470²,
(1 + 2 + ... + 15)(16)(17 + 18 + ... + 28) = 720²,
(1 + 2 + ... + 15)(16 + 17 + ... + 23)(24 + 25 + ... + 28) = 1560²,
(1 + 2 + ... + 17)(18 + 19 + ... + 22)(23 + 24 + ... + 28) = 1530².

1³ + 2³ + ... + 28³ = (1 + 2 + ... + 28)² = 406².

28² = 784 appears in the decimal expressions of π and e:
  π = 3.14159•••784••• (from the 1795th digit),
  e = 2.71828•••784••• (from the 436th digit), (784 is the 6th 3-digit square in the expr. of e).

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Page of Squares : First Upload November 3, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan