The square of 5.
The smallest squares containing k 25's :
25 = 52, 225625 = 4752, 25250625 = 50252,
25832525625 = 1607252, 6252525255025 = 25005052.
The squares which begin with 25 and end in 25 are
255025 = 5052, 2512225 = 15852, 2544025 = 15952, 2576025 = 16052,
25050025 = 50052,...
The first integer which is the sum of 10 squares in just two ways (see 20).
The 2nd integer which is the sum of 7 squares in just two ways (see 24).
252 = 625 is a zigzag square.
252 = (12 + 4)(112 + 4).
252 = 625 is an exchangeable square : 256 = 162.
252 is the first square which is the sum of 2 squares in just 2 ways: 72 + 242 = 152 + 202.
252 is the sum of 5 fourth powers : 24 + 24 + 34 + 44 + 44.
25 is contained in its square (625).
The alternating sum of the first 25 cubics is a square: 13 - 23 + 33 - ... + 253 = 912.
258 = 152587890625,
1 + 5 + 2 + 587 + 8 + 9 + 06 + 2 + 5 = 1 + 525 + 8 + 7 + 8 + 9 + 062 + 5 = 252,
2511 = 2384185791015625, 2 + 3 + 8 + 4 + 1 + 8 + 5 + 7 + 9 + 10 + 1 + 562 + 5 = 252 and more 31 equations.
(252 + 5) = (12 + 5)(102 + 5) = (42 + 5)(52 + 5),
(252 - 5) = (52 - 5 )(62 - 5).
Komachi Fractions : (25/26)2 = 84375/91260, (25/33)2 = 16875/29403.
Komachi equations:
252 = 12 * 345 / 6 + 7 - 8 * 9 = 12 * 345 / 6 - 7 * 8 - 9
= 1 * 234 + 56 * 7 + 8 - 9, and more 33 equations,
252 = 9 * 8 * 7 + 6 * 5 * 4 + 3 - 2 * 1 = 987 - 6 * 5 * 4 * 3 - 2 * 1
= 9 + 8 * 76 + 5 + 4 - 3 + 2 * 1, and more 42 equations,
252 = 98 * 7 - 6 + 5 - 4 * 3 / 2 * 10 = 9 + 8 * 76 - 5 + 4 - 3 + 2 + 10
= 9 + 8 * 76 - 5 - 4 - 3 + 2 * 10, and more 28 equations,
252 = 12 * 22 - 32 + 42 * 52 + 62 + 72 + 82 + 92 = 12 * 232 + 42 - 562 / 72 / 82 + 92
= 12 - 22 * 342 + 562 / 72 + 82 * 92,
252 = 92 + 82 + 72 + 62 + 52 * 42 - 32 + 22 * 12 = 92 + 82 + 72 + 62 + 52 * 42 - 32 + 22 / 12.
252 + 262 + 272 + ... + 482 = 1822,
252 + 262 + 272 + ... + 502 = 1952,
252 + 262 + 272 + ... + 732 = 3572,
252 + 262 + 272 + ... + 5782 = 80332,
252 + 262 + 272 + ... + 6242 = 90102,
252 + 262 + 272 + ... + 36252 = 1260352,
252 + 262 + 272 + ... + 216242 = 18359402.
(1)(2 + 3 + ... + 25) = 182,
(1 + 2 + ... + 8)(9 + 10 + ... + 25) = 1022,
(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + ... + 25) = 1982,
(1 + 2 + 3)(4 + 5 + ... + 24)(25) = 2102,
(1 + 2 + ... + 7)(8 + 9 + ... + 16)(17 + 18 + ... + 25) = 7562.
13 + 23 + 33+ 43 + ... + 253 = (1 + 2 + 3 + 4 + ... + 25)2 = 3252,
(13 + 23 + ... + 83)(93 + 103 + ... + 253) = 116282,
(13 + 23 + ... + 133)(143 + 153 + ... + 253) = 283922,
(13 + 23 + ... + 223)(233 + 243 + 253) = 516122.
252 = 625 appears in the decimal expressions of π and e:
π = 3.14159•••625••• (from the 1568th digit),
e = 2.71828•••625••• (from the 590th digit), (625 is the 9th 3-digit square in the expr. of e).
Page of Squares : First Upload October 20, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan