The smallest squares containing k 12's :
121 = 112, 1201216 = 10962, 121242121 = 110112,
1121212030129 = 10588732, 812121128112121 = 284977392.
The third integer which is the sum of a square and a prime in just 2 ways : 12 + 11, 32 + 3.
122 is the third squares which is the sum of 5 distinct squares: 12 + 22 + 32 + 72 + 92.
122 is the 4th squares which is the sum of 3 squares in just one way: 42 + 82 + 82.
122 is the 4th squares which is the sum of 4 squares in just 2 ways:
22 + 22 + 62 + 102 = 62 + 62 + 62 + 62.
122 is the 5th squares which is the sum of 4 cubes: 23 + 23 + 43 + 43.
12! + 1 is divisible by 132.
122 = 4! + 5!
122 = 144, 1 + 4 + 4 = 32.
1 + 22 + 3 + 42 + 5 + 62 + 7 + 82 + 9 + 102 + 11 + 122 = 202.
15 + 24 + 35 + 44 + 55 + 64 + 75 + 84 + 95 + 104 + 115 + 124 = 5262.
6k + 12k + 36k + 129k + 258k are squares for k= 1,2,3,4 (see the page of 6).
122 = 144 is a reversible square (441 = 212).
122 = 144, where every digit is a square.
125 = 248832, 24 + 88 + 32 = 122,
(122+ 6) = (22+ 6)(32+ 6),
(122- 8) = (42- 8)(52- 8) = (32- 8)(42- 8)(52- 8).
122 = 2 x 3 x 4 + 4 x 5 x 6.
122 = (1)(2)(3 + 4 + 5)(6).
122± 5 are primes.
122 = (22 - 1)(72 - 1) = (13 + 8)(23 + 8).
122 + 132 + 142 + ... + 502 = 512 + 522 + 532 + ... + 632.
Komachi Fractions: 122 = 278640/1935 = 387504/2691
(12/13)2 = 42768/50193, (12/19)2 = 20736/51984
Komachi equations:
122 = 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 103 equations,
122 = 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 132 equations,
122 = 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 99 equations,
122 = 12 + 22 * 342 + 52 - 672 + 82 - 92 = - 122 + 342 - 52 * 62 + 72 + 82 - 92
= 12 + 22 + 32 * 42 + 52 + 62 - 72 + 82 - 92, and more 4 equations,
122 = 92 - 82 - 72 * 62 + 542 - 322 - 12 = - 92 - 82 + 72 + 652 - 42 - 32 * 212
= 92 + 82 + 72 - 62 - 52 + 42 - 32 + 22 * 12, and more 7 equations,
122 = 982 / 72 - 62 - 52 / 42 * 322 / 102 = 92 + 82 + 72 + 62 + 52 - 42 + 32 - 22 - 102
= 92 - 82 + 72 - 62 + 52 - 42 + 32 - 22 + 102, and more 3 equations,
122 = 123 + 33 + 43 + 53 - 63 - 73 - 83 - 93 = 123 - 33 - 43 - 53 + 63 - 73 - 83 - 93
= 13 - 23 * 33 + 43 + 563 / 73 + 83 - 93,
122 = 93 + 83 + 73 - 63 - 53 - 43 - 33 - 23 - 103 = - 93 + 83 - 73 - 63 - 53 + 43 - 33 + 23 + 103,
122 = - 94 - 84 + 74 - 64 - 54 + 44 + 34 - 24 + 104
(1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12) = 152,
(1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + 11 + 12) = 502,
(1)(2 + 3)(4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)(12) = 602,
(1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12) = 902,
(1 + 2)(3)(4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12) = 1802,
(1 + 2 + 3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11 + 12) = 2102,
(1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12) = 2702,
(1)(2 + 3)(4 + 5 + 6)(7)(8)(9 + 10 + 11 + 12) = 4202,
(1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9)(10 + 11)(12) = 4202,
(1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11)(12) = 5402,
(1 + 2)(3 + 4)(5)(6)(7 + 8)(9 + 10 + 11 + 12) = 6302,
(1 + 2 + 3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11)(12) = 7202,
(1 + 2 + 3 + 4)(5)(6)(7)(8)(9 + 10 + 11 + 12) = 8402,
(1)(2 + 3)(4)(5 + 6)(7 + 8)(9)(10 + 11 + 12) = 9902,
(1 + 2)(3 + 4 + 5)(6)(7 + 8)(9 + 10 + 11)(12) = 10802,
(1 + 2)(3)(4 + 5)(6)(7 + 8)(9 + 10 + 11)(12) = 16202,
(1)(2 + 3)(4)(5)(6)(7 + 8)(9 + 10 + 11)(12) = 18002,
(1)(2 + 3)(4)(5 + 6)(7 + 8 + 9)(10)(11)(12) = 26402,
(1)(2)(3 + 4)(5)(6)(7 + 8)(9)(10 + 11)(12) = 37802.
(12)(22 + 32 + 42 + 52)(62 + 72)(82 + 92 + 102 + 112 + 122) = 15302.
13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 + 113 + 123 = (1 + 2 + 3 + ... + 12)2,
(13 + 23 + 33 + 43)(53 + 63 + 73 + 83)(93 + 103)(113 + 123) = 7953402,
(13 + 23 + 33)(43)(53 + 63 + 73 + 83)(93 + 103)(113 + 123) = 38176322,
(13)(23)(33)(43 + 53 + 63)(73 + 83 + 93)(103)(113)(123) = 5645376002,
(13 + 23 + 33 + 43)(53)(63)(73 + 83 + 93)(103)(113)(123) = 31363200002,
(13 + 23 + 33)(43)(53)(63)(73 + 83 + 93)(103)(113)(123) = 150543360002.
125 = 248832, 242 + 82 + 82 + 322 = 123,
126 = 2985984, 29 + 8 + 5 + 98 + 4 = 122,
127 = 35831808, 35 + 83 + 18 + 08 = 122,
128 = 429981696, 4 + 2 + 9 + 9 + 8 + 16 + 96 = 4 + 2 + 9 + 98 + 16 + 9 + 6
= 4 + 2 + 99 + 8 + 16 + 9 + 6 = 4 + 29 + 9 + 81 + 6 + 9 + 6
= 42 + 9 + 9 + 8 + 1 + 69 + 6 = 122,
129 = 5159780352, 5 + 1 + 5 + 9 + 7 + 80 + 35 + 2 = 5 + 15 + 9 + 78 + 035 + 2 = 122.
1210 = 61917364224, 6 + 1 + 9 + 1 + 73 + 6 + 42 + 2 + 4 = 122 and more 6 equations,
1211 = 743008370688, 7 + 43 + 008 + 3 + 7 + 068 + 8 = 74 + 3 + 008 + 37 + 06 + 8 + 8
74 + 30 + 08 + 3 + 7 + 06 + 8 + 8 = 122,
1212 = 8916100448256, 8 + 9 + 1 + 6 + 10 + 04 + 48 + 2 + 56 = 122 and more 9 equations,
1213 = 106993205379072, 1 + 06 + 9 + 9 + 3 + 2 + 05 + 3 + 7 + 90 + 7 + 2 = 122 and more 8 equations,
1214 = 1283918464548864, 1 + 2 + 8 + 3 + 9 + 18 + 4 + 6 + 4 + 5 + 4 + 8 + 8 + 64 = 122 and more 11 equations.
122 = 144 appears in the decimal expressions of π and e:
π = 3.14159•••144••• (from the 1638th digit),
e = 2.71828•••144••• (from the 2070th digit).
Page of Squares : First Upload September 8, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan