The smallest squares containing k 24's :
324 = 182, 1249924 = 11182, 2424249 = 15572,
242426247424 = 4923682, 2424472424500624 = 492389322.
The squares which begin with 24 and end in 24 are
2458624 = 15682, 24186724 = 49182, 24324624 = 49322, 24681024 = 49682,
24820324 = 49822,...
The first integer which is the sum of 9 squares in just two ways (see 20).
The second integer which is the sum of 6 squares in just two ways:
24 = 12 + 12 + 12 + 12 + 22 + 42 = 22 + 22 + 22 + 22 + 22 + 22.
The first square which is the sum of 6 cubes in just 3 ways:
13 + 13 + 23 + 33 + 33 + 83 = 23 + 23 + 43 + 43 + 63 + 63 = 33 + 33 + 33 + 33 + 53 + 73.
242 = 43 + 83.
242 = 576, 5 + 76 = 92.
242 = 576 is a zigzag square.
242 = (22 - 1)(32 - 1)(52 - 1).
24 and 25 are consecutive integers having square factors (the second case).
52 + 24 = 72, 52 - 24 = 12.
2458624 = 15682.
247 = 4586471424, 4 + 5 + 8 + 64 + 71 + 424 = 4 + 5 + 86 + 471 + 4 + 2 + 4
= 45 + 8 + 6 + 471 + 42 + 4 = 458 + 64 + 7 + 1 + 42 + 4 = 242,
248 = 110075314176, 1 + 10 + 07 + 531 + 4 + 17 + 6 = 242 and more 5 equations,
249 = 2641807540224, 2 + 6 + 4 + 1 + 8 + 07 + 540 + 2 + 2 + 4 = 242 and more 4 equations,
2410 = 63403380965376, 6 + 3 + 4 + 03 + 3 + 80 + 96 + 5 + 376 = 242 and more 16 equations,
2411 = 1521681143169024, 1 + 5 + 2 + 1 + 6 + 8 + 1 + 1 + 431 + 6 + 90 + 24 = 242 and more 33 equations.
(242 - 2) = (22 - 2)(172 - 2), (242 + 4) = (42 + 4)(52 + 4), (242 + 9) = (22 + 9)(62 + 9)
(242 - 6) = (32 - 6)(142 - 6) = (52 - 6)(62 - 6) = (32 - 6)(42 - 6)(52 - 6).
12 + 22 + 32+ 42+ ... + 242 = 702,
13 + 23 + 33+ 43+ ... + 243 = 3002.
(1 + 2 + 3)(4 + 5 + ... + 24) = 422.
22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).
Komachi Fraction : (24/259)2 = 5184/603729
Komachi equations:
242 = 12 / 3 * 45 * 6 - 7 * 8 * 9 = 123 - 45 - 6 + 7 * 8 * 9
= 1 * 2 * 345 - 6 * 7 - 8 * 9, and more 48 equations,
242 = 9 + 8 + 7 * 6 * 5 * 4 / 3 * 2 - 1 = 98 * 7 / 6 * 5 + 4 / 3 + 2 + 1
= 9 * 8 * 7 + 65 + 4 * 3 / 2 + 1, and more 41 equations,
242 = 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 21 equations,
242 = 12 - 232 / 42 + 52 * 62 * 72 / 82 - 92,
242 = 92 - 82 * 72 + 62 + 52 * 42 * 32 - 22 - 12,
242 = 123 + 33 + 43 + 53 + 63 - 73 - 83 - 93 = 13 + 23 * 33 + 43 + 563 / 73 + 83 - 93
= - 13 - 23 * 33 + 43 - 563 + 73 * 83 + 93, and more 8 equations,
242 = 93 + 83 + 73 + 63 - 53 - 43 - 33 - 23 - 103 = 93 + 83 + 73 - 63 + 53 + 43 + 33 - 23 - 103
= - 93 + 83 - 73 + 63 - 53 + 43 - 33 + 23 + 103,
242 = 14 * 24 + 34 - 44 - 54 + 64 - 74 - 84 + 94 = 94 - 84 - 74 + 64 - 54 - 44 + 34 + 24 * 14
= 94 - 84 - 74 + 64 - 54 - 44 + 34 + 24 / 14.
The sum of the squares of the divisors of 24 is equal to the sum of the squares of the divisors of 26.
(12 + 22)(32 + 4~2 + ... + 72)(82 + 92 + ... + 122)(132 + 142 + ... + 242) = 382502,
(12 + 22 + 32 + 42)(52 + 62 + ... + 92)(102 + 112)(122 + 132 + ... + 242) = 861902,
(12 + 22 + ... + 102)(112 + 122)(132)(142 + 152 + ... + 242) = 2652652.
15572= 2424249.
242 = 576 appears in the decimal expressions of π and e:
π = 3.14159•••576••• (from the 2004th digit),
e = 2.71828•••576••• (from the 520th digit), (576 is the 8th 3-digit square in the expr. of e).
Page of Squares : First Upload October 20, 2003 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan