The smallest squares containing k 35's :
4356 = 66²,   13535041 = 3679²,   26351353561 = 162331²,
24353573584356 = 4934934²,   1357357353513561 = 36842331².

The 6th integer which is the sum of 3 distinct squares : 1² + 3² + 5².

The 3rd integer which is the sum of 5 squares in just 3 ways.
The 1st integer which is the sum of 8 squares in just 3 ways (see 32).

1 + 2 + 3 + ... + 49 = 35².

35 = (1² + 2² + 3² + ... + 24²) / (1² + 2² + 3² + ... + 7²).

35² is the alternating sum of 49 consecutive squares : 1² - 2² + 3² - 4² + ... + 49².
35² is the alternating sum of 13 consecutive cubes: 1³ - 2³ + 3³ - 4³ + ... + 13³.

35² is the sum of the cubes of consecutive odd integers : 1³ + 3³ + 5³ + 7³ + 9³.

35² = (1² + 6)(13² + 6).

35² = 1³ + 2³ + 6³ + 10³ = 3³ + 7³ + 7³ + 8³ = 4³ + 6³ + 6³ + 9³.

35² = 1225, 1 = 1², 225 = 15².

35⁸ = 2251875390625, 225 + 1 + 8 + 7 + 53 + 906 + 25 = 35²,
35⁸ = 2251875390625, 225 + 1 + 8 + 75 + 3 + 906 + 2 + 5 = 35²,
35⁸ = 2251875390625, 225 + 1 + 875 + 3 + 90 + 6 + 25 = 35².

Cubic Polynomial (X + 35²)(X + 72²)(X + 96²) = X³ + 125²X² + 8088²X + 241920².

20^k + 260^k + 265^k + 680^k are squares for k = 1,2,3 (35², 775²,18725²).
81^k + 136^k + 304^k + 704^k are squares for k = 1,2,3 (35², 783², 19495²).
217^k + 218^k + 272^k + 518^k are squares for k = 1,2,3 (35²,661²,13405²).
240^k + 306^k + 313^k + 366^k are squares for k = 1,2,3 (35²,619²,11053²).

Komachi Fraction : 35² = 1069425/873.

Komachi equations:
35² = 1234 + 5 - 6 - 7 + 8 - 9 = 1234 - 5 + 6 + 7 - 8 - 9
  = 1 + 2 * 3 * 4 * 5 * 6 + 7 * 8 * 9, and more 9 equations,
35² = 98 + 7 * 6 + 543 * 2 - 1 = 98 * 7 - 6 + 543 + 2 * 1
  = 9 * 8 * 7 + 6 * 5 * 4 * 3 * 2 + 1, and more 4 equations,
35² = 987 + 654 / 3 + 2 * 10 = 9 + 876 + 5 * 4 + 32 * 10
  = 9 + 8 - 7 - 65 + 4 * 32 * 10, and more 4 equations,
35² = 1² - 2² + 3² - 4² * 5² + 6² * 7² - 8² - 9² = 12² * 3² - 4² - 5² + 6² - 7² + 8² - 9²,
35² = 9² - 8² + 7² + 6² * 5² * 4² / 3² - 21² = 98² / 7² - 6² + 5² + 4² + 32² * 1²
  = 98² / 7² - 6² + 5² + 4² + 32² / 1².

(35² + 5) = (5² + 5)(6² + 5),   (35² - 7) = (6² - 7)(7² - 7).

1³ + 2³ + ... + 35³ = (1 + 2 + ... + 35)² = 630²,
(1³ + 2³ + ... + 27³)(28³ + 29³ + ... + 35³) = 190512².

35² = 1225 appears in the decimal expressions of π and e:
  π = 3.14159•••1225••• (from the 9416th digit),
  e = 2.71828•••1225••• (from the 8977th digit).

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Page of Squares : First Upload November 24, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan