The smallest squares containingk 68's :
1681 = 412, 1868689 = 13672, 686806849 = 262072,
3656856868681 = 19122912, 1568268268268689 = 396013672.
682 = 4624, 4 + 6 + 2 + 4 = 42.
682 = 4624, a zigzag square with even digits.
68 is the sum of m squares for m = 2,3,....,54.
682 = 23 + 23 + 83 + 163 = 23 + 83 + 93 + 153 = 33 + 53 + 123 + 143 = 24 + 44 + 44 + 84.
688 = 457163239653376,
4 + 571 + 63 + 2 + 3965 + 3 + 3 + 7 + 6 = 4571 + 6 + 3 + 2 + 3 + 9 + 6 + 5 + 3 + 3 + 7 + 6
= 4 + 571 + 6 + 32 + 3965 + 3 + 37 + 6 = 4 + 571 + 6 + 32 + 3965 + 33 + 7 + 6 = 682
Komachi equations:
682 = - 12 - 3 + 4567 + 8 * 9 = 9 / 8 * 76 * 54 + 3 * 2 + 1
= 9 * 87 + 6 * 5 * 4 * 32 + 1 = - 9 - 8 + 7 * 654 + 3 * 21
= 9 * 8 + 76 * 5 * 4 * 3 + 2 - 10 = - 9 + 876 * 5 + 43 + 210,
682 = 92 + 82 + 762 - 542 / 32 * 22 - 12 = 92 * 82 + 72 - 652 + 42 + 32 * 22 * 102
= - 92 - 82 * 72 + 652 + 42 + 32 * 22 * 102.
The third integer which is the sum of a square and a prime in 4 ways :
12 + 67, 32 + 59, 52 + 43, 72 + 19.
(682 - 4) = (82 - 4)(92 - 4) = (32 - 4)(42 - 4)(92 - 4)
(12 + 22 + ... + 252)(262 + 262 + ... + 512)(522 + 532 + ... + 592)(602 + 612 + ... + 682) = 4488112202.
682 = 4624 appears in the decimal expressions of π and e:
π = 3.14159•••4624••• (from the 5877th digit),
e = 2.71828•••4624••• (from the 2028th digit).
Page of Squares : First Upload February 9, 2004 ; Last Revised March 26, 2010
by Yoshio Mimura, Kobe, Japan