The smallest squares containingk 85's :
28561 = 169²,   85285225 = 9235²,   3698585856 = 60816²,
15858585856656 = 3982284²,   14385407858585856 = 119939184².

The third integer which is the sum of 2 distinct squares in 2 ways : 2² + 9² = 6² + 7².

The alternating sum of the first 85 cubes is a square, 559².

1² + 2² + 3² + ... + 85² = 1 + 2 + 3 + ... + 645 = 208335,
the 4th triangle number which is the sum of the squares of consecutive integers.
(Such integers are 1, 55, 91, 208335, ...)

85² = 7225, 7 + 2 + 2 + 5 = 4²,   7³ + 2³ + 2³ + 5³ = 22².

85² = 1! + 4! + 6! + 6! + 6! + 7!.

85² = (2² + 1)(38² + 1).

85², 132², 720², 85² + 132² = 157², 132² + 720² = 732², 720² + 85² = 725².

85⁵ = 4437053125, 4 + 43 + 7053 + 125 = 85²,
85⁵ = 4437053125, 44 + 3 + 7053 + 125 = 85².

A 3-by-3 magic square consisting of different squares with constant 85²:

408^k + 1156^k + 2601^k + 3060^k are squares for k = 1,2,3 (85², 4199², 218773²).
85^k + 1649^k + 4097^k + 4573^k are squares for k = 1,2,3 (102², 6358², 410958²).

Komachi equations:
85² = 9 - 8 + 7 * 6 / 5 * 43 * 2 * 10 = 98 * 76 - 5 * 43 + 2 - 10,
85² = 1² + 2² + 34² + 5² + 6² + 78² - 9².

(85² - 5) = (9² - 5)(10² - 5).

(1 + 2 + ... + 21)(22)(23 + 24 + ... + 85) = 4158²,
(1 + 2 + ... + 49)(50 + 51 + ... + 58)(59 + 60 + ... + 85) = 34020².

(1²)(2² + 3² + ... + 9²)(10² + 11² + ... + 14²)(15² + 16² + ... + 85²) = 207320²,
(1² + 2² + ... + 25²)(26²)(27² + 28² + ... + 59²)(60² + 61² + ... + 85²) = 181717250².

85² = 7225 appears in the decimal expressions of π and e:
  π = 3.14159•••7225••• (from the 8421st digit),
  e = 2.71828•••7255••• (from the 3780th digit).

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Page of Squares : First Upload March 8, 2004 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan