The smallest squares containing k 59's :
5929 = 772, 595984 = 7722, 59159859984 = 2432282,
1595959735969 = 12633132, 4595959591359049 = 677935072.
The 3rd integer which is the sum of 5 squares in just 5 ways.
The 2nd integer which is the sum of 8 squares in just 8 ways.
The 1st integer which is the sum of 11 squares in just 10 ways.
1 / 59 = 0.0169...., 169 = 132.
592 is the 7th square which is the sum of 3 cubes : 592 = 23 + 93 + 143.
592 = 3481 is a zigzag square whose digits are distinct.
592 = 3481, 3 + 4 + 8 + 1 = 42,
592 = 3481, 3 + 481 = 222.
Komachi equations:
592 = 1 / 2 * 34 * 5 * 6 * 7 - 89 = - 1 + 2 + 3456 + 7 + 8 + 9
= 9 + 8 * 76 * 5 + 432 */ 1 = 9 + 87 * 6 * 5 * 4 / 3 + 2 - 10.
592 = 12 / 22 * 32 * 42 + 52 - 62 * 72 + 82 * 92 = 122 - 32 + 42 + 562 + 72 + 82 + 92
= 12 + 22 + 342 + 562 * 72 / 82 - 92 = 92 * 82 - 72 * 62 + 52 + 42 * 32 / 22 */ 12
= 92 * 82 * 72 / 62 + 52 - 42 * 32 / 22 * 102 = 982 / 72 * 62 + 52 - 42 * 32 / 22 * 102
= 92 + 82 * 72 + 62 + 542 / 32 + 22 - 102 = 92 - 82 + 72 * 62 + 52 / 42 * 322 + 102
= 982 / 72 + 652 - 42 - 322 + 102 = - 982 / 72 + 62 + 52 + 42 + 32 * 22 * 102.
(592 + 3) = (72 + 3)(82 + 3), (592 + 5) = (12 + 5)(242 + 5).
2532 = 272 + 282 + 292 + ... + 592.
(1 + 2)(3)(4 + 5 + ... + 59) = 1262,
(1 + 2)(3 + 4 + ... + 21)(22 + 23 + ... + 59) = 10262,
(1 + 2)(3 + 4 + ... + 42)(43 + 44 + ... + 59) = 15302,
(1 + 2 + ... + 8)(9 + 10 + ... + 42)(43 + 44 + ... + 59) = 52022,
(1 + 2 + ... + 32)(33 + 34 + ... + 48)(49 + 50 + ... + 59) = 142562,
(1 + 2 + ... + 38)(39 + 40 + ... + 57)(58 + 59) = 88922.
(12)(22 + 32 + ... + 102)(112 + 122 + ... + 152)(162 + 172 + ... + 592) = 1504802,
(12 + 22)(32 + 42 + ... + 62)(72 + 82 + ... + 552)(562 + 572 + ... + 592) = 5688902,
(12 + 22 + ... + 52)(62 + 72 + ... + 102)(112 + 122 + ... + 152)(162 + 172 + ... + 592) = 10345502,
(12 + 22 + ... + 82)(92 + 102 + ... + 202)(212 + 222 + ... + 512)(522 + 532 + ... + 592) = 239300162,
(12 + 22 + ... + 92)(102 + 112 + ... + 172)(182 + 192 + ... + 342)(352 + 362 + ... + 592) = 169575002,
(12 + 22 + ... + 172)(182 + 192 + ... + 342)(352 + 362 + ... + 422)(432 + 442 + ... + 592) = 1062075002,
(12 + 22 + ... + 242)(252)(262)(272 + 282 + ... + 592) = 115115002.
592 = 3481 appears in the decimal expressions of π and e:
π = 3.14159•••3481••• (from the 31339th digit),
e = 2.71828•••3481••• (from the 3440th digit).
Page of Squares : First Upload January 26, 2004 ; Last Revised March 19, 2010
by Yoshio Mimura, Kobe, Japan