The smallest squares containing k 29's :
529 = 23²,   29929 = 173²,   1654292929 = 40673²,
2929299129 = 54123²,   292935992929129 = 17115373².

The squares which begin with 29 and end in 29 are
29929 = 173²,   2968729 = 1723²,   2982529 = 1727²,   29408929 = 5423²,
29452329 = 5427²,...

The first integer which is the sum of 5 squares in just 3 ways (see 28)
The 3rd integer which is the sum of 6 squares in just 2 ways.
The 3rd integer which is the sum of 8 squares in just 2 ways (see 28).
The 4th integer which is the sum of 3 distinct squares: 2² + 3² + 4².

29 = (1²)(2² + 3² + 4²).

29² = 1! + 5! + 6!.

29² = 3⁰ + 3¹ + 3³ + 3⁴ + 3⁶.

29^k + 109^k + 301^k + 461^k are squares for k = 1,2,3 (30², 562², 11250²).
62^k + 73^k + 266^k + 440^k are squares for k = 1,2,3 (29², 523², 10229²).

Komachi Fractions : (4/29)² = 9648/507123, (13/29)² = 10647/52984,
(29/38)² = 30276/51984, (29/213)² = 7569/408321, (29/217)² = 7569/423801.

Komachi equations:
29² = 1 + 2 - 3 + 4 * 5 * 6 * 7 - 8 + 9 = 1 - 2 + 3 + 4 * 5 * 6 * 7 + 8 - 9
  = 12 * 3 * 4 * 5 / 6 * 7 - 8 + 9, and more 14 equations,
29² = 9 - 8 + 7 * 6 * 5 * 4 + 3 - 2 - 1 = 9 - 8 + 7 * 6 * 5 * 4 - 3 + 2 + 1
  = 9 * 87 + 65 - 4 * 3 / 2 - 1, and more 22 equations,
29² = 9 + 8 + 7 * 6 * 5 * 4 - 3 * 2 - 10 = 9 * 87 + 6 + 5 * 4 * 3 + 2 - 10
  = 9 * 87 + 65 + 4 - 3 + 2 - 10, and more 15 equations,
29² = 1² + 2² / 3² * 45² + 6² + 7² - 8² - 9²,
29² = 9² * 8² - 76² + 5² + 43² - 21².

29² = 841, 8² + 4² + 1² = 9²,
29² = 841, 8 + 41 = 7².

29² = 841 with a decreasing sequence of digits 8, 4, 1.

29² = 841, where 81 = 9², 4 = 2², the second mosaic square.

2x² + 29 are prime for x = 0, 1, 2,..., 28

29² is the sum of 2 consecutive squares: 20² + 21².

29⁸ = 500246412961, 500 + 246 + 4 + 1 + 29 + 61 = 29².

(29² - 1) = (5² - 1)(6² - 1) = (2² - 1)(3² - 1)(6² - 1),
(29² - 5) = (4² - 5)(9² - 5),   (29² + 9) = (4² + 9)(5² + 9).

7² + 8² + 9² + ... + 29² = 92².

(1 + 2)(3 + 4 + ... + 29) = 36²,
(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + ... + 29) = 945²,
(1 + 2 + ... + 13)(14 + 15 + ... + 22)(23 + ... + 29) = 1638²,
(1 + 2 + ... + 14)(15 + 16 + ... + 20)(21 + ... + 29) = 1575².

(1³ + 2³ + ... + 24³)(25³ + 26³ + ... + 29³) = 94500²,
(1² + 2² + ... + 13²)(14²)(15² + 16² + ... + 27²)(28² + 29²)= 1242150².

1³ + 2³ + ... + 29³ = (1 + 2 + ... + 29)² = 435².

1² + 2² + 3² + ... + 29² = 8555, which consists of only 2 kinds of digits.

29² = 841 appears in the decimal expressions of π and e:
  π = 3.14159•••841••• (from the 35th digit), (841 is the first 3-digit square in the expr. of π)
  e = 2.71828•••841••• (from the 230th digit), (841 is the 4th 3-digit square in the expr. of e).

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Page of Squares : First Upload November 3, 2003 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan