The smallest squares containing k 20's :
2025 = 452, 205209 = 4532, 2015920201 = 448992,
9320522020209 = 30529532, 202207120202025 = 142199552.
The first integer which is the sum of 5 squares in just two ways:
20 = 12 + 12 + 12 + 12 + 42 = 22 + 22 + 22 + 22 + 22.
Every integer greater than 20 is the sum of 7 nonzero squares.
20 = (12 + 22 + 32 + ... + 1042) / (12 + 22 + 32 + ... + 382).
20k + 260k + 265k + 680k are squares for k = 1,2,3 (352, 7752, 187252).
Komachi equations:
202 = 123 * 4 * 5 / 6 + 7 - 8 - 9 = 1 + 2 * 34 * 5 + 6 * 7 + 8 + 9
= 1 + 2 * 34 * 5 - 6 + 7 * 8 + 9, and more 36 equations,
202 = 9 + 8 + 76 * 5 + 4 - 3 + 2 * 1 = 9 + 8 + 76 * 5 - 4 + 3 * 2 + 1
= 9 + 8 - 7 + 65 * 4 * 3 / 2 * 1, and more 29 equations,
202 = 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 33 equations,
202 = 12 + 22 - 32 + 42 * 52 + 62 - 72 - 82 + 92 = 12 - 22 / 32 + 42 * 52 / 62 * 72 - 82 - 92
= - 12 - 22 + 32 + 42 * 52 - 62 + 72 + 82 - 92 = - 12 + 22 - 32 + 452 - 62 * 72 + 82 + 92,
202 = 92 * 872 * 62 / 542 / 32 - 212 = 92 + 82 * 72 - 652 + 432 - 212
= 92 - 82 - 72 + 62 + 52 * 42 - 32 + 22 + 12, and more 4 equations,
202 = 92 * 82 / 72 / 62 / 542 * 32 * 2102 = - 92 - 82 + 72 / 62 * 542 / 32 + 22 + 102
= - 92 + 82 * 72 + 62 - 542 + 32 / 22 * 102,
202 = 94 + 84 - 74 - 64 - 544 / 34 / 24 + 14 = - 94 + 84 - 74 - 64 + 544 / 34 / 24 + 14.
The sum of the squares of the divisors of 202 is 312, a square.
12 + 22 + 32 + ... + 202 = 2870, which consists of different digits.
202 = 1 + 7 + 72 + 73.
We consider the following process: Starting with an integer, form a new integer by adding the squares of its digits. Repeat these steps :
20 - 4 - 16 - 37 - 58 - 89 - 145 - 42 - 20 (the starting integer)
(202 + 8) = (32 + 8)(42 + 8).
202 + 212 = 292,
202 + 212 + 222 + ... + 432 = 1582,
202 + 212 + 222 + ... + 3082 = 31282.
13 + 23 + 33 + 43 + ... + 203 = (1 + 2 + 3 + 4 + ... + 20)2 = 2102,
202 = (1)(2)(3 + 4 + 5 + 6 + 7)(8).
202 = 4 x 5 + 6 x 7 + 8 x 9 + 10 x 11 + 12 x 13,
202 = 5 x 6 + 6 x 7 + 7 x 8 + 8 x 9 + 9 x 10 + 10 x 11.
(1 + 2 + ... + 6)(7 + 8 + ... + 20) = 632,
(1 + 2 + ... + 14)(15 + 16 + ... + 20) = 1052.
(12 + 22 + 32)(42 + 52 + ... + 102)(112 + 122 + ... + 142)(152 + 162 + ... + 202) = 779102,
(12 + 22 + ... + 72)(82 + 92 + 102)(112 + 122)(132)(142)(152 + 162 + ... + 202) = 236327002,
(12 + 22)(32)(42 + 52 + ... + 102)(112)(122)(132)(142)(152 + 162 + ... + 202) = 1336935602,
(12 + 22 + 32)(42)(52 + 62 + 72)(82 + 92 + 102)(112 + 122 + 132)(142)(152 + 162 + 172)(182 + 192 + 202) = 6549928002,
(12 + 22)(32)(42 + 52 + 62)(72)(82 + 92 + 102)(112 + 122 + 132)(142)(152 + 162 + 172)(182 + 192 + 202) = 17193561002,
(12)(22)(32)(42 + 52 + 62)(72)(82)(92)(102)(112 + 122 + 132)(142)(152 + 162 + 172)(182 + 192 + 202) = 707392224002,
(12)(22)(32 + 42 + 52)(62)(72)(82 + 92 + 102)(112 + 122 + 132)(142)(152)(162)(172)(182 + 192 + 202) = 3644141760002,
(12)(22 + 32 + 42 + 52 + 62)(72)(82)(92)(102)(112 + 122 + 132)(142)(152)(162)(172)(182 + 192 + 202) = 18741300480002,
(12 + 22)(32 + 42 + 52)(62)(72)(82)(92)(102)(112 + 122 + 132)(142)(152)(162)(172)(182 + 192 + 202) = 187413004800002.
(13)(23 + 33 + 43 + 53)(63 + 73 + 83 + 93 + 103)(113)(123)(133 + 143 + 153 + 163 + 173 + 183 + 193 + 203) = 2341785602, and more 103 equations.
(15)(25 + 35)(45)(55)(65)(75)(85)(95)(105 + 115)(125 + 135 + 145 + 155 + 165 + 175 + 185 + 195)(205) = 403341431856168960000002,
(15 + 25)(35)(45)(55)(65)(75)(85)(95)(105 + 115)(125 + 135 + 145 + 155 + 165 + 175 + 185 + 195)(205) = 2178043732023312384000002.
202 = 400 appears in the decimal expressions of π and e:
π = 3.14159•••400••• (from the 1174th digit),
e = 2.71828•••400••• (from the 1595th digit).
Page of Squares : First Upload October 6, 2003 ; Last Revised January 25, 2011
by Yoshio Mimura, Kobe, Japan