The smallest squares containing k 11's :
1156 = 342, 2114116 = 14542, 1101111489 = 331832,
11116116119056 = 33340842, 111111711230116 = 105409542.
The second integer which is the sum of a square and a prime in just 2 ways : 22 + 7, 32 + 2.
The 4th integer which is the sum of 3 squares: 12 + 12 + 32.
The 3rd integer which is the sum of 5 squares: 12 + 12 + 12 + 22 + 22.
112 is the third square which is the sum of 3 distinct squares: 22 + 62 + 92.
112 is the second square which is the sum of 4 distinct squares: 12 + 22 + 42 + 102.
112 is the second square which is the sum of 5 distinct squares: 12 + 22 + 42 + 62 + 82.
112 is the first square which is the sum of 3 squares in 2 ways: 22 + 62 + 92 = 62 + 62 + 72.
112 is the first square which is the sum of 4 squares in 3 ways:
12 + 22 + 42 + 102 = 22 + 22 + 72 + 82 = 42 + 42 + 52 + 82.
112 = 121 is a palindromic square (of a palindromic number 11), and a zigzag square.
112 = 121, 1 + 2 + 1 = 22.
112 = 5! + 1.
12 + 22 + 32 + ... + 112 = 506, 5 + 0 + 6 = 11.
112 = 30 + 31 + 32 + 33 + 34.
112 and 22 are the only squares which become cubes when increased by 4.
2x2 + 11 are prime for x = 0,1,2,..,10
112 = 121, and 12 - 1 = 11.
(12 + 22 + 32 + 42) + (12 + 22 + 32 + 42 + 52 + 62) = 112.
118 = 214358881, 2 + 1 + 43 + 58 + 8 + 8 + 1 = 2 + 14 + 3 + 5 + 8 + 8 + 81
= 2 + 14 + 3 + 5 + 8 + 88 + 1 = 2 + 14 + 3 + 5 + 88 + 8 + 1 = 112,
1114 = 379749833583241, 3 + 7 + 9 + 7 + 4 + 9 + 8 + 3 + 3 + 58 + 3 + 2 + 4 + 1
= 3 + 7 + 9 + 7 + 4 + 9 + 8 + 3 + 35 + 8 + 3 + 24 + 1
= 3 + 7 + 9 + 7 + 4 + 9 + 8 + 33 + 5 + 8 + 3 + 24 + 1
= 37 + 9 + 7 + 4 + 9 + 8 + 3 + 3 + 5 + 8 + 3 + 24 + 1 = 112.
(112 - 1) = (32 - 1)(42 - 1),
(112 + 5) = (22 + 5)(32 + 5) = (12 + 5)(42 + 5),
(112 + 7) = (12 + 7)(32 + 7), (112 - 7) = (32 - 7)(82 - 7),
(112 + 9) = (12 + 9)(22 + 9), (112 - 9) = (42 - 9)(52 - 9).
Komachi Fractions: (7/11)2 = 15876/39204, (8/11)2 = 19584/37026 = 74052/39168,
(11/13)2 = 67518/94302, (11/15)2 = 20691/38475 = 41382/76950,
(11/148)2 = 3267/591408
Komachi equations:
112 = 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 133 equations,
112 = 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 185 equations,
112 = 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 136 equations,
112 = 12 * 22 / 32 * 452 / 62 - 72 + 82 + 92 = - 12 * 22 + 32 + 452 - 62 * 72 - 82 - 92
= 12 / 22 * 32 * 42 + 52 - 62 - 72 + 82 + 92, and more 5 equations,
112 = 92 - 82 - 72 + 62 - 542 / 32 + 212 = 92 + 82 + 72 - 62 - 52 - 42 + 32 - 22 - 12
= 92 + 82 - 72 + 62 + 52 - 42 * 32 / 22 * 12, and more 8 equations,
112 = 982 / 72 + 62 + 52 - 42 * 32 / 22 - 102 = 982 / 72 + 62 * 52 / 42 / 32 * 22 - 102
= 92 - 82 + 72 + 62 - 542 / 32 / 22 + 102, and more 4 equations,
112 = 13 - 23 - 33 - 43 - 53 - 63 + 73 - 83 + 93,
112 = 93 - 83 + 73 - 63 - 53 - 43 - 33 - 23 + 13.
112 + 122 + 132 + ... + 229082 = 20018632.
(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11) = 282,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11) = 302,
(1)(2)(3 + 4 + 5 + 6 + 7 + 8 + 9)(10 + 11) = 422,
(1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11) = 902,
(1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + 10 + 11) = (1)(2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11) = 1802,
(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11) = (1)(2)(3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11) = 1982,
(1)(2)(3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11) = (1)(2)(3 + 4)(5)(6 + 7 + 8 + 9)(10 + 11) = 2102,
(1 + 2)(3 + 4 + 5)(6 + 7 + 8)(9)(10 + 11) = 3782,
(1 + 2)(3)(4 + 5)(6 + 7 + 8)(9)(10 + 11) = 5672,
(1)(2 + 3)(4)(5)(6 + 7 + 8)(9)(10 + 11) = 6302,
(1 + 2)(3 + 4)(5 + 6)(7)(8 + 9 + 10)(11) = 6932,
(1 + 2)(3 + 4)(5 + 6 + 7)(8)(9)(10 + 11) = (1 + 2)(3 + 4 + 5 + 6)(7)(8)(9)(10 + 11) = 7562,
(1)(2)(3 + 4)(5)(6)(7)(8)(9 + 10 + 11) = 8402,
(1 + 2)(3 + 4 + 5)(6)(7)(8)(9)(10 + 11) = 15122,
(1 + 2 + 3)(4)(5 + 6)(7 + 8)(9)(10)(11) = (1)(2)(3)(4)(5 + 6)(7 + 8)(9)(10)(11) = 19802,
(1 + 2)(3)(4 + 5)(6)(7)(8)(9)(10 + 11) = 22682,
(1)(2 + 3)(4)(5)(6)(7)(8)(9)(10 + 11) = 25202.
13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 + 113 = (1 + 2 + ... + 11)2 = 662,
(13)(23 + 33 + 43)(53)(63 + 73 + 83 + 93)(103)(113) = 54450002,
(13)(23 + 33)(43 + 53)(63)(73 + 83 + 93)(103)(113) = 548856002,
(13 + 23)(33)(43)(53)(63)(73 + 83 + 93)(103)(113) = 9408960002.
112 = 121 appears in the decimal expressions of π and e:
π = 3.14159•••121••• (from the 710th digit), 121 is the 5th 3-digit square in the expr. of π,
e = 2.71828•••121••• (from the 1027th digit),
Page of Squares : First Upload September 1, 2003 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan