The smallest squares containingk 60's :
1600 = 402, 4609609 = 21472, 216060601 = 146992,
2056060606609 = 14338972, 602946060601600 = 245549602.
The 4th integer which is the sum of 4 squares in just 3 ways:
60 = 12 + 12 + 32 + 72 = 12 + 32 + 52 + 52= 22 + 22 + 42 + 62.
The first integer which is the sum of 6 squares in just 7 ways:
[1,1,1,2,2,7], [1,1,1,4,4,5], [1,1,2,2,5,5], [1,1,2,3,3,6], [1,3,3,3,4,4], [2,2,2,4,4,4], [2,2,3,3,3,5].
602 = 5 x 6!
602 = (22 - 1)(42 - 1)(92 - 1).
602 = (1)(2)(3)(4)(5)(6 + 7 + 8 + 9) = (1)(2)(3 + 4 + 5 + 6 + 7)(8)(9)
= (1)(2 + 3)(4)(5 + 6 + ... + 19) = (1)(2 + 3)(4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)(12)
= (1 + 2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10 + ... + 28)
= (1 + 2 + 3 + 4 + ... + 8)(9 + 10 + 11 + 12 + ... + 16) = (1 + 2 + 3)(4)(5)(6 + 7 + 8+ 9).
602 + 612 + 622 + ... + 1102 = 1112 + 1122 + 1132 + ... + 1352.
602 = 13 + 23 + 63 + 153 = 13 + 73 + 83 + 143 = 23 + 43 + 113 + 133.
9k + 10k + 60k + 90k are squares for k = 1,2,3 (132, 1092, 9732).
60k + 241k + 282k + 378k are squares for k = 1,2,3 (312, 5332, 95212).
Komachi equations:
602 = 12 * 3 / 4 * 56 * 7 + 8 * 9 = 12 - 3 + 456 * 7 / 8 * 9
= - 1 - 2 - 3 * 45 + 6 * 7 * 89 = 9 + 876 + 543 / 2 * 10
= 9 * 8 * 7 * 6 - 54 + 3 * 210 = - 9 + 8 * 7 * 65 - 43 + 2 + 10,
602 = 12 * 22 + 32 * 42 * 52 - 62 + 72 + 82 - 92 = 122 * 32 - 42 + 562 * 72 / 82 - 92
= 122 * 32 - 42 * 52 + 62 * 782 / 92 = 12 - 232 - 42 + 52 / 62 * 782 - 92
= - 12 * 22 + 32 * 42 * 52 + 62 - 72 - 82 + 92 = - 12 - 22 * 32 * 42 + 562 / 72 * 82 + 92
= 92 - 82 - 72 + 62 + 52 * 42 * 32 - 22 */ 12 = 92 - 82 + 72 * 62 - 52 + 432 - 22 - 12
= 92 * 82 + 72 * 62 - 542 + 32 - 212 = 92 * 872 * 62 / 542 - 32 * 212
= - 92 + 82 + 72 - 62 + 52 * 42 * 32 + 22 */ 12 = 982 - 762 - 542 / 32 - 22 + 102.
(12 + 22 + 32 + ... + 602) = 73810, which consists of different digits.
(602 + 4) = (72 + 4)(82 + 4).
602 + 612 + 622 + ... + 32382 = 1064032,
602 + 612 + 622 + ... + 922 = 4402.
12 + 22 + 32 + 42 + ... + 602 = 73810, which consists of different digits.
(1 + 2 + 3 + 4)(5 + 6 + ... + 20)(21 + 22 + ... + 60) = 18002,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 60) = 1652,
(1 + 2 + ... + 8)(9 + 10 + ... + 14)(15 + 16 + ... + 60) = 20702.
(12 + 22 + ... + 202)(212)(222 + 232 + ... + 552)(562 + 572 + ... + 602) = 338114702.
602 = 3600 appears in the decimal expressions of π and e:
π = 3.14159•••3600••• (from the 358th digit),
(3136 is the second 4-digit square in the expression of π,)
e = 2.71828•••3600••• (from the 6740th digit).
Page of Squares : First Upload February 2, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan