The smallest squares containingk 61's :
361 = 19²,   136161 = 369²,   1570616161 = 39631²,
61216161561 = 247419²,   61166179616161 = 7820881².

The squares which begin with 61 and end in 61 are
6155361 = 2481²,   61136761 = 7819²,   61324561 = 7831²,   61921161 = 7869²,
611028961 = 24719²,...

61 is the sum of m squares for m = 2,3,...,47.

61² = 3721, a square with different digits.

61² = 1! + 5! + 6! + 6! + 6! + 6! + 6!.

6155361 = 2481².

The alternating sum of the first 61 cubes is a square : 1³ - 2³ + 3³ - .. +61³ = 341².

Komachi equations:
61² = 1 * 2 * 34 * 56 - 78 - 9 = 12 - 34 + 5 + 6 * 7 * 89
  = 9 + 87 / 6 / 5 * 4 * 32 * 10 = 9 + 8 - 76 + 54 / 3 * 210
  = 9 * 8 * 7 + 6 + 5 - 4 + 3210 = 9 + 87 * 6 - 5 * 4 + 3210
  = - 9 - 8 - 7 * 6 + 54 / 3 * 210 = - 9 - 8 * 7 + 6 + 54 / 3 * 210
  = - 9 * 8 + 7 + 6 + 54 / 3 * 210,
61² = - 1² - 23² + 4² - 5² * 6² - 7² + 8² * 9² = - 98² + 7² + 6² * 5² * 4² - 32² - 10²,
61² = 9³ + 87³ / 6³ - 5³ + 4³ + 3³ / 2³ + 1³,
61² = - 1⁵ - 2⁵ + 3⁵ - 4⁵ - 56⁵ / 7⁵ - 8⁵ + 9⁵.

(61² + 3) = (2² + 3)(23² + 3) = (2² + 3)(4² + 3)(5² + 3),
(61² - 3) = (4² - 3)(17² - 3),
(61² + 5) = (7² + 5)(8² + 5) = (1² + 5)(2² + 5)(8² + 5).

(1² + 2² + 3² + ... + 61²) = 77531, which consists of odd digits (the 5th 5-digit sum).
The first 5-digit integer whose digits are non-increasing.

(1)(2 + 3 + ... + 19)(20 + 21 + ... + 61) = 567²,
(1)(2 + 3 + ... + 43)(44 + 45 + ... + 61) = 945²,
(1 + 2 + 3)(4 + 5 + ... + 16)(17 + 18 + ... + 61) = 1170²,
(1 + 2 + ... + 6)(7 + 8 + ... + 19)(20 + ... + 61) = 2457²,
(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + ... + 61) = 2583²,
(1 + 2 + ... + 6)(7 + 8 + ... + 28)(29 + ... + 61) = 3465²,
(1 + 2 + ... + 6)(7 + 8 + ... + 57)(58 + ... + 61) = 2856²,
(1 + 2 + ... + 7)(8 + 9 + ... + 34)(35 + ... + 61) = 4536²,
(1 + 2 + ... + 13)(14 + 15 + ... + 22)(23 + ... + 61) = 4914²,
(1 + 2 + ... + 14)(15 + 16 + ... + 20)(21 + ... + 61) = 4305²,
(1 + 2 + ... + 14)(15 + 16 + ... + 33)(34 + ... + 61) = 7980²,
(1 + 2 + ... + 14)(15 + 16 + ... + 43)(44 + ... + 61) = 9135²,
(1 + 2 + ... + 15)(16 + 17 + ... + 19)(20 + ... + 61) = 3780²,
(1 + 2 + ... + 15)(16 + 17 + ... + 50)(51 + ... + 61) = 9240²,
(1 + 2 + ... + 26)(27 + 28 + 29)(30 + ... + 61) = 6552²,
(1 + 2 + ... + 27)(28 + 29 + ... + 57)(58 + ... + 61) = 10710²,
(1 + 2 + ... + 33)(34 + 35 + ... + 50)(51 + ... + 61) = 15708²,
(1 + 2 + ... + 38)(39 + 40 + ... + 52)(53 + ... + 61) = 15561²,
(1 + 2 + ... + 45)(46)(47 + ... + 61) = 6210²,
(1 + 2 + ... + 50)(51 + 52 + ... + 57)(58 + 59 + ... + 61) = 10710².

(1²)(2² + 3² + ... + 6²)(7² + 8² + ... + 61²) = 2640²,
(1² + 2² + ... + 25²)(26² + 27² + ... + 50²)(51² + 52² + ... + 61²) = 2674100²,
(1² + 2²)(3² + 4² + 5²)(6²)(7² + 8² + ... + 61²) = 26400².

61² = 3721 appears in the decimal expressions of π and e:
  π = 3.14159•••3721••• (from the 5978th digit),
  e = 2.71828•••3721••• (from the 1273rd digit).

---

Page of Squares : First Upload February 2, 2004 ; Last Revised March 26, 2010
by Yoshio Mimura, Kobe, Japan