The smallest squares containing k 52's :
529 = 232, 528529 = 7272, 45252529 = 67272,
3952052552529 = 19879772, 5225235291352521 = 722857892.
The first integer which is the sum of 4 squares in just 4 ways:
[1,1,1,7], [1,1,5,5], [2,4,4,4], [3,3,3,5]
The first integer which is the sum of 13 squares in just 7 ways.
The first integer which is the sum of 10 seventh powers.
The square root of 52 is 7.2111..., 7 = 22 + 12 + 12 + 12.
522 = 2704 is a zigzag square whose digits are distinct.
(12 + 22 + 32 + ... + 522) = 48230, which consists of different digits.
52k + 91k + 117k + 169k + 247k are squares for k = 1,2,3,4. (262, 3382 ,47322, 692902).
528 = 53459728531456, 5 + 3 + 459 + 728 + 53 + 1456 = 522,
528 = 53459728531456, 53 + 459 + 728 + 5 + 3 + 1456 = 522.
52k + 190k + 233k + 254k are squares for k = 1,2,3 (272, 3972, 60032).
Komachi equations:
522 = 1 + 2 + 3 * 4 + 5 * 67 * 8 + 9 = 1 - 2 + 34 + 5 * 67 * 8 - 9
= 1 + 23 + 45 * 67 * 8 / 9 = 1 - 23 * 45 + 6 * 7 * 89
= 9 + 87 * 6 * 5 + 43 * 2 - 1 = 9 * 8 * 7 * 6 + 5 - 4 - 321
= - 9 * 8 - 7 + 65 * 43 - 2 - 10,
522 = 12 * 22 + 32 - 42 * 52 + 62 + 72 * 82 - 92 = - 122 / 32 + 452 - 672 + 82 * 92
= 92 * 82 * 72 * 62 / 542 + 32 - 212 = - 92 + 82 * 72 + 62 - 52 * 42 + 32 + 22 * 12
= - 92 + 82 * 72 + 62 - 52 * 42 + 32 + 22 / 12 = - 982 + 72 - 652 + 42 * 322 + 102,
522 = 93 + 83 + 73 + 63 - 53 + 43 - 33 - 23 + 103.
(522 + 2) = (12 + 2)(302 + 2), (522 - 7) = (62 - 7)(102 - 7),
(522 - 4) = (72 - 4)(82 - 4) = (32 - 4)(42 - 4)(72 - 4).
522 + 532 + 542 + ... + 1472 = 10122,
522 + 532 + 542 + ... + 3892 = 44332.
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 34)(35 + 36 + ... + 52) = 26102,
(1 + 2 + ... + 10)(11 + 12 + ... + 19)(20 + 21 + ... + 52) = 29702,
(1 + 2 + ... + 25)(26 + 27 + ... + 52) = 5852.
12 + 22 + 32 + 42 + ... + 522 = 48230, which consists of different digits.
(12 + 22 + ... + 62)(72 + 82 + ... + 522) = 20932,
(12 + 22 + 32)(42 + 52 + ... + 382)(392 + 402 + ... + 482)(492 + 502 + 512 + 522) = 71838902,
(12 + ... + 82)(92 + 102 + ... + 172)(182 + 192 + 202)(212 + 222 + ... + 522) = 39841202.
522 = 2704 appears in the decimal expressions of π and e:
π = 3.14159•••2704••• (from the 2668th digit),
e = 2.71828•••2704••• (from the 815th digit),
(2704 is the seventh 4-digit square in the expression of e.)
Page of Squares : First Upload January 19, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan