The smallest squares containing k 18's :
1849 = 432, 183184 = 4282, 181818256 = 134842,
1831818781809 = 13534472, 1803185181189184 = 424639282.
182 = 324, a zigzag squares.
182 = 324, 3 + 2 + 4 = 32,
182 = 324, 32 + 4 = 62.
182 + 192 + 202 + ... + 282 = 772.
182 + 192 + 202 + ... + 342 = 352 + 362 + 372 + ... + 422.
182 is the sum of m squares for m = 3,4,5,6,7,8.
182 is the second square which is the sum of 4 squares in just 4 ways.
182 = 34 + 34 + 34 + 34.
10k + 74k + 98k + 142k are squares for k = 1,2,3 (182, 1882, 20522).
14k + 58k + 122k + 130k are squares for k = 1,2,3 (182, 1882, 20522).
34k + 82k + 98k + 110k are squares for k = 1,2,3 (182, 1722, 16922).
18k + 84k + 129k + 210k are squares for k = 1,2,3 (212, 2612, 34652).
18k + 282k + 921k + 1380k are squares for k = 1,2,3 (512, 16832, 585812).
Komachi Fraction: 182 = 486972/1503.
Komachi equations:
182 = 1 + 2 + 3 - 4 - 5 + 6 * 7 * 8 - 9 = 1 - 2 - 3 - 4 + 5 + 6 * 7 * 8 - 9
= 1 * 2 - 3 - 4 * 5 + 6 * 7 * 8 + 9, and more 40 equations,
182 = 9 + 8 * 7 * 6 - 5 * 4 - 3 + 2 * 1 = 9 * 8 * 7 - 6 * 5 * 4 * 3 / 2 * 1
= 98 * 7 - 6 * 5 * 4 * 3 - 2 * 1, and more 52 equations,
182 = 9 + 8 * 7 * 6 - 5 - 4 * 3 / 2 - 10 = 9 * 8 * 7 - 6 * 5 * 4 - 3 * 2 * 10
= 98 + 7 * 6 * 5 + 4 * 3 / 2 + 10, and more 44 equations,
182 = 122 + 32 * 42 * 52 + 62 * 72 - 82 * 92 = 12 - 232 - 42 + 52 * 62 - 72 - 82 + 92
= 12 - 22 + 32 * 42 + 52 - 62 + 72 + 82 + 92, and more 3 equations,
182 = 982 / 72 / 62 * 542 * 32 / 212 = 92 + 82 * 72 + 62 - 542 - 32 - 22 * 12
= 92 + 82 + 72 - 62 + 52 + 42 * 32 - 22 + 12, and more 8 equations,
182 = 92 + 82 - 72 - 62 + 52 * 42 - 32 * 22 - 102 = 92 - 82 - 72 + 62 * 52 - 42 * 32 - 22 * 102
182 = 982 - 762 - 52 * 42 * 32 - 22 + 102,
182 = - 13 * 23 - 33 + 43 + 563 / 73 + 83 - 93.
(182 + 1) = (22 + 1)(82 + 1), (182 + 6) = (32 + 6)(42 + 6)
(182 - 2) = (42 - 2)(52 - 2) = (22 - 2)(32 - 2)(52 - 2).
182 + 192 + 202 + ... + 282 = 772.
12 + 22 + 32 + ... + 182 = 2109, which consists of different digits.
18 = (12 + 22 + 32 + ... + 272) / (12 + 22 + 32 + ... + 102).
182 = (1)(2)(3)(4 + 5)(6) = (1 + 2 + 3)(4 + 5)(6) = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)(9)
= (1)(2 + 3 + ... + 25).
182 = (12 + 2)(22 + 2)(42 + 2) = (12 + 5)(72 + 5).
(1 + 2)(3 + 4 + ... + 6)(7 + 8 + ... + 18) = 902,
(1 + 2)(3 + 4 + ... + 17)(18) = 902,
(1 + 2)(3 + 4 + ... + 9)(10 + 11 + ... + 18) = 1262,
(1 + 2 + 3)(4 + 5 + ... + 17)(18) = 1262,
(1 + 2 + 3)(4 + 5 + ... + 14)(15 + 16 + ... + 18) = 1982,
(1 + 2 + ... + 16)(17)(18) = 2042,
(1 + 2 + ... + 4)(5 + 6 + ... + 9)(10 + 11 + ... + 18) = 2102,
(1 + 2 + ... + 4)(5 + 6 + ... + 16)(17 + 18) = 2102,
(1 + 2 + ... + 6)(7 + 8 + 9)(10 + 11 + ... + 18) = 2522,
(1 + 2 + ... + 6)(7 + 8 + ... + 11)(12 + 13 + ... + 18) = 3152,
(1 + 2 + ... + 9)(10 + 11)(12 + 13 + ... + 18) = 3152,
(13 + 23 + ... + 183) = (1 + 2 + ... + 18)2 = 1712,
(products of at most 3 factors)
188 = 11019960576, 1 + 1 + 0199 + 60 + 57 + 6 = 1 + 101 + 99 + 60 + 57 + 6 = 182,
189 = 198359290368, 1 + 9 + 83 + 5 + 92 + 90 + 36 + 8 and more 5 equations,
1810 = 3570467226624,
3 + 5 + 7 + 04 + 6 + 7 + 2 + 266 + 24 = 3 + 5 + 7 + 04 + 6 + 7 + 226 + 62 + 4
= 3 + 5 + 7 + 04 + 67 + 226 + 6 + 2 + 4 = 3 + 5 + 7 + 046 + 7 + 226 + 6 + 24 = 182.
1811 = 64268410079232, 6 + 4 + 26 + 84 + 100 + 7 + 92 + 3 + 2 = 182 and more 8 equations,
1812 = 1156831381426176, 1 + 1 + 5 + 6 + 8 + 3 + 1 + 3 + 8 + 14 + 261 + 7 + 6 = 182 and more 126 equations.
In the set {18, 882, 2482, 4743} the sum of any two numbers is a square :
18 + 882 = 302, 18 + 2482 = 502, 18 + 4743 = 692,
882 + 2482 = 582, 882 + 4743 = 752, 2482 + 4743 = 852
182 = 324 appears in the decimal expressions of π and e:
π = 3.14159•••324••• (from the 2494th digit),
e = 2.71828•••324••• (from the 795th digit).
Page of Squares : First Upload September 29, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan