The smallest squares containingk 69's :
169 = 132, 69169 = 2632, 4690469169 = 684872,
69926926969 = 2644372, 169697856936969 = 130268132.
The squares which begin with 69 and end in 69 are
69169 = 2632, 6953769 = 26372, 69105969 = 83132, 69505569 = 83372,
69939769 = 83632,...
1 / 69 = 0.0144..., 144 = 122.
69 = (12 + 22 + 32 + ... + 222) / (12 + 22 + 32 + 42 + 52).
692 is the second square which is the sum of 8 seventh powers : [1,1,1,2,2,2,3,3].
692 = 4761 with different digits.
692 = 4761, 4 + 76 + 1 = 92.
692 = 232 + 462 + 462, 642 + 642 + 322 = 962.
692 = 73 + 73 + 113 + 143.
69169 = 2632, a square,
69696 = 2642, a square.
Every digit appears in either 692 = 4761 or 693 = 328509.
Komachi equations:
692 = 1 * 234 + 567 * 8 - 9 = 1 + 2 * 3 + 4 * 5 + 6 * 789
= 12 * 3 - 4 - 5 + 6 * 789 = - 1 - 2 - 3 * 4 * 5 + 67 * 8 * 9
= - 1 * 2 * 34 + 5 + 67 * 8 * 9 = - 1 * 2 + 34 - 5 + 6 * 789
= - 12 + 34 + 5 + 6 * 789 = 9 * 87 * 6 + 5 * 4 * 3 + 2 + 1
= 9 + 87 * 65 - 43 * 21 = 9 * 87 * 6 + 54 - 3 + 2 + 10
= 987 - 6 + 54 / 3 * 210,
692 = - 12 * 22 * 32 - 42 * 52 - 62 + 72 + 82 * 92 = 92 * 82 + 72 - 62 - 52 * 42 - 32 * 22 */ 12
= 92 - 82 + 762 - 52 + 42 - 322 + 12 = - 982 - 72 + 62 * 52 * 42 + 32 + 22 + 12
= 92 + 82 + 72 * 62 + 542 + 32 * 22 - 102 = 92 - 82 + 72 * 62 + 542 - 32 * 22 + 102
= 982 - 762 + 52 - 42 + 322 - 102 = - 92 + 872 - 62 - 542 + 32 / 22 * 102.
(692 - 3) = (82 - 3)(92 - 3) = (32 - 3)(42 - 3)(82 - 3),
(692 + 9) = (32 + 9)(162 + 9).
(1 + 2)(3 + 4 + ... + 51)(52 + 53 + ... + 69) = 20792,
(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 69) = 3602,
(1 + 2 + ... + 4)(5 + 6 + ... + 20)(21 + 22 + ... + 69) = 21002,
(1 + 2 + ... + 6)(7 + 8 + ... + 42)(43 + 44 + ... + 69) = 52922,
(1 + 2 + ... + 12)(13 + 14 + ... + 51)(52 + 53 + ... + 69) = 102962,
(1 + 2 + ... + 15)(16 + 17 + ... + 65)(66 + 67 + ... + 69) = 81002,
(1 + 2 + ... + 26)(27 + 28 + ... + 51)(52 + 53 + ... + 69) = 193052,
(1 + 2 + ... + 32)(33 + 34 + ... + 66)(67 + 68 + 69) = 134642,
(1 + 2 + ... + 48)(49 + 50 + 51)(52 + 53 + ... + 69) = 138602,
(1 + 2 + ... + 49)(50 + 51 + ... + 55)(56 + 57 + ... + 69) = 183752,
(1 + 2 + ... + 50)(51)(52 + 53 + ... + 69) = 84152.
(13 + 23 + ... + 43)(53)(63 + 73 + ... + 693) = 2700002.
697 = 7446353252589, 74 + 4635 + 3 + 2 + 5 + 25 + 8 + 9 = 692,
697 = 7446353252589, 74 + 4635 + 3 + 25 + 2 + 5 + 8 + 9 = 692,
698 = 513798374428641, 5 + 1 + 3 + 79 + 8 + 374 + 4286 + 4 + 1 = 692,
5 + 1 + 37 + 9 + 8 + 374 + 4286 + 41 = 5 + 1 + 379 + 8 + 3 + 74 + 4286 + 4 + 1
= 5 + 1 + 379 + 8 + 37 + 4 + 4286 + 41 = 51 + 3798 + 37 + 4 + 4 + 2 + 864 + 1 = 692,
699 = 35452087835576229, 3 + 5 + 4520 + 8 + 7 + 83 + 5 + 57 + 62 + 2 + 9 = 692,
3+5+4520+8+7+83+55+7+62+2+9 = 3+5+4520+8+78+3+55+76+2+2+9
= 3+5+4520+8+78+35+5+76+2+29 = 3+5+4520+8+78+35+5+76+22+9
= 3+5+4520+87+8+3+5+57+62+2+9 = 3+5+4520+87+8+3+55+7+62+2+9
= 3+5+4520+87+8+35+5+7+62+29 = 3+545+2+08+7+8+3557+622+9
= 35+4+52+0878+3557+6+229 = 35+452+08+78+3557+622+9
= 35+4520+8+7+8+35+57+62+29 = 35+4520+8+7+83+5+5+7+62+29
= 35+4520+8+78+3+5+5+76+2+29 = 35+4520+8+78+3+5+5+76+22+9
= 35+4520+8+78+35+5+7+62+2+9 = 35+4520+87+8+3+5+5+7+62+29
= 35+4520+87+8+35+57+6+2+2+9 = 35+4520+87+83+5+5+7+6+2+2+9
= 354+520+87+8+3557+6+229 = 3545+2+08+7+8+3+557+622+9
= 3545+2+08+78+355+762+2+9 = 3545+2+087+835+57+6+229
= 3545+20+878+3+5+5+76+229 = 3545+208+7+8+355+7+622+9 = 692.
692 = 4761 appears in the decimal expressions of π and e:
π = 3.14159•••4761••• (from the 26057th digit),
e = 2.71828•••4761••• (from the 240th digit),
(4761 is the second 4-digit square in the expression of e.)
Page of Squares : First Upload February 9, 2004 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan