The smallest squares containing k 21's :
121 = 112, 212521 = 4612, 121242121 = 110112,
21319212121 = 1460112, 52121021421121 = 72194892.
The squares which begin with 21 and end in 21 are
212521 = 4612, 2134521 = 14612, 21058921 = 45892, 21261321 = 46112,
21520321 = 46392,...
The first integer which is the sum of 6 squares in just two ways (see 20).
The 2nd integer which is the sum of 3 distinct squares: 12 + 22 + 42.
1 / 21 = 0.04761..., 4761 = 692.
212 = 441, 4 + 4 + 1 = 32.
212 = 13 + 23 + 63 + 63 = 13 + 23 + 33 + 43 + 53 + 63.
212± 2 are primes (the 4th case).
Every integer greater than 21 is the sum of 8 nonzero squares.
212 = 441 is a reversible square (144 = 122).
212 = 441, every digit of which is a square.
212 + 222 + 232 + 242 = 252 + 262 + 272.
18k + 84k + 129k + 210k are squares for k = 1,2,3 (212, 2612, 34652).
21k + 45k + 201k + 633k are squares for k = 1,2,3 (302, 6662, 161822).
Komachi Fractions : (2/21)2 = 8460/932715, (21/95)2 = 7938/162450.
Komachi equations:
212 = 1 + 2 - 3 * 4 * 5 - 6 + 7 * 8 * 9 = 1 * 2 + 3 * 4 * 5 * 6 + 7 + 8 * 9
= 123 - 4 - 5 + 6 * 7 * 8 - 9, and more 44 equations,
212 = 9 * 8 + 7 + 6 * 5 * 4 * 3 + 2 * 1 = 9 * 8 * 7 - 6 - 5 * 4 * 3 + 2 + 1
= 9 * 8 + 76 * 5 - 4 - 3 * 2 - 1, and more 44 equations,
212 = 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 31 equations,
212 = 12 + 22 + 342 - 52 + 672 - 82 * 92 = 12 * 22 / 32 * 42 * 5672 / 82 / 92
= 12 + 22 * 32 + 42 * 52 + 62 - 72 - 82 + 92, and more 3 equations,
212 = 982 / 72 / 62 * 542 / 32 / 22 * 12 = 92 * 82 - 762 + 52 - 42 + 322 * 12
= 92 - 82 - 72 + 62 + 52 * 42 + 32 * 22 + 12, and more 4 equations,
212 = 92 * 872 * 62 / 542 / 32 - 22 * 102 = 92 + 82 * 72 - 652 + 432 - 22 * 102
= 982 / 72 - 62 + 52 - 42 * 32 + 22 * 102, and more 6 equations,
212 = 13 + 23 + 33 - 43 + 53 - 63 + 73 - 83 + 93 = 13 * 23 + 33 / 43 * 563 / 73 - 83 + 93,
212 = 93 - 83 + 73 - 63 + 53 - 43 + 33 + 23 + 13.
218 = 37822859361, 3 + 78 + 2 + 285 + 9 + 3 + 61 = 378 + 2 + 2 + 8 + 5 + 9 + 36 + 1 = 212,
219 = 794280046581, 7 + 94 + 280 + 046 + 5 + 8 + 1 = 79 + 4 + 280 + 04 + 65 + 8 + 1 = 212,
2110 = 16679880978201, 16 + 6 + 7 + 98 + 8 + 097 + 8 + 201 = 212 and more 7 equations,
2111 = 350277500542221, 3 + 50 + 277 + 50 + 054 + 2 + 2 + 2 + 1 = 212 and more 9 equations,
2112 = 7355827511386641, 7 + 3 + 5 + 5 + 8 + 2 + 7 + 5 + 1 + 1 + 386 + 6 + 4 + 1 = 212 and more 35 equations.
(212 + 1) = (42 + 1)(52 + 1), (212 + 7) = (12 + 7)(72 + 7),
(212 + 9) = (12 + 9)(62 + 9) = (32 + 9)(42 + 9),
(212 - 9) = (52 - 9)(62 - 9).
292 = 202 + 212 + 222 + ... + 212.
212 + 222 + 232 + ... + 1162 = 7242.
212 = (1 + 2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10),
212 = (1 + 2 + 3 + 4 + 5 + 6)2 = 13 + 23 + 33 + 43 + 53 + 63.
(1 + 2)(3)(4 + 5 + ... + 21) = 452,
(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 21) = 1082,
(1 + 2 + ... + 8)(9 + 10 + ... + 18)(19 + 20 + 21) = 5402.
13 + 23 + ... + 213 = (1 + 2 + ... + 21)2 = 2312,
(13)(23 + 33 + ... + 133)(143 + 153 + ... + 213) = 193202,
(13 + 23 + ... + 63)(73 + 83 + ... + 143)(153 + 163 + ... + 213) = 4445282.
12 + 22 + 32 + ... + 212 = 3311, which consists of 2 kinds of odd digits (the first 4-digit sum).
212 = 441 appears in the decimal expressions of π and e:
π = 3.14159•••441••• (from the 726th digit), (441 is the 6th 3-digit square in the expr. of π,)
e = 2.71828•••441••• (from the 1288th digit).
Page of Squares : First Upload October 6, 2003 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan