The smallest squares containing k 17's :
1764 = 422, 11771761 = 34312, 3171717124 = 563182,
1717171747281 = 13104092, 173217817178176 = 131612242.
17 is the 4th square which is the sum of 2 distinct squares: 12 + 42.
172 = 289 with an increasing sequence of digits: 2, 8, 9
172 is the sum of m distinct squares for m = 4,5,6,7,8.
172 is the 5th square which is the sum of 2 distinct squares: 82 + 152.
172 is the second square which is the sum of 3 squares in just 2 ways:
12 + 122 + 122 = 82 + 92 + 122.
172 is the first square which is the sum of 4 squares in just 7 ways, and the first square which is the sum of 4 distinct squares in just 4 ways.
172 is the second square which is the sum of 4 distinct cubes: 13 + 23 + 43 + 63.
172 is the 4th square which is the sum of 4 fourth powers: 14 + 24 + 24 + 44.
172 - 1 = 2(4! + 5!).
17k + 91k + 299k + 377k are squares for k = 1,2,3 (282, 4902, 90042).
Komachi Fractions: 172 = 920754/3186, (14/17)2 = 15876/23409
Komachi equations:
172 = 1 * 2 / 3 / 4 * 5 * 6 * 7 * 8 + 9 = 1 / 2 * 3 * 4 * 5 / 6 * 7 * 8 + 9
= 12 * 34 - 5 - 6 * 7 - 8 * 9, and more 47 equations,
172 = 9 + 8 * 7 * 6 * 5 / 4 / 3 * 2 * 1 = 9 + 8 * 7 / 6 * 5 * 4 * 3 / 2 * 1
= 9 * 8 + 7 * 6 * 5 + 4 * 3 / 2 + 1, and more 41 equations,
172 = 9 + 8 + 7 * 6 * 5 * 4 / 3 + 2 - 10 = 9 - 8 + 7 * 6 * 5 * 4 / 3 - 2 + 10
= 9 * 8 + 7 - 6 * 5 + 4 * 3 * 2 * 10, and more 57 equations,
172 = 12 * 22 * 32 / 42 * 562 / 72 + 82 + 92 = 122 * 32 - 42 * 562 / 72 - 82 + 92
= 12 + 22 - 32 * 42 * 52 / 62 * 72 + 82 * 92, and more 4 equations,
172 = 92 * 82 - 72 / 62 * 52 * 42 * 32 + 22 + 12 = - 92 + 82 - 72 + 62 + 542 / 32 - 22 - 12
= - 92 + 82 - 72 - 62 - 52 - 42 - 32 + 212,
172 = 92 + 82 + 72 * 62 * 52 + 42 * 32 - 2102 = 92 + 82 + 72 * 62 * 52 * 42 * 32 / 2102
= 92 + 82 - 72 * 62 * 52 + 42 * 32 + 2102 = - 92 - 82 - 72 - 62 - 52 + 42 * 32 + 22 * 102.
(12 + 22 + 33) + (42 + 52 + 63) = 172
(172 + 1) = (12 + 1)(122 + 1), (172 - 3) = (42 - 3)(52 - 3),
(172 + 5) = (32 + 5)(42 + 5), (172 + 8) = (12 + 8)(52 + 8),
(172 - 9) = (42 - 9)(72 - 9).
172 + 182 + 192 + ... + 392 = 1382,
172 + 182 + 192 + ... + 53452 = 2256432.
12 + 22 + 32 + ... + 172 = 1785, which consists of different digits.
(1 + 2 + ... + 4)(5 + 6 + 7)(8 + 9 + ... + 17) = 1502,
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 17) = 1802,
(1 + 2 + ... + 4)(5 + 6 + ... + 10)(11 + 12 + ... + 17) = 2102,
(1 + 2 + ... + 9)(10)(11 + 12 + ... + 17) = 2102,
(1 + 2 + ... + 9)(10 + 11 + ... + 14)(15 + 16 + 17) = 3602,
(13 + 23 + ... + 173) = (1 + 2 + ... + 17)2 = 1532,
(products of at most 3 factors)
1710 = 2015993900449,
201 + 5 + 9 + 9 + 3 + 9 + 004 + 49 = 201 + 5 + 9 + 9 + 3 + 9 + 0044 + 9 = 172.
1712 = 582622237229761,
5 + 8 + 2 + 6 + 2 + 2 + 237 + 2 + 2 + 9 + 7 + 6 + 1 = 172 and more 40 equations.
172 = 289 appears in the decimal expressions of π and e:
π = 3.14159•••289••• (from the 1310th digit),
e = 2.71828•••289••• (from the 1399th digit).
Page of Squares : First Upload September 22, 2003 ; Last Revised January 25, 2011
by Yoshio Mimura, Kobe, Japan