The square of 8.
The smallest squares containingk 64's :
64 = 82, 166464 = 4082, 416486464 = 204082,
64649164644 = 2542622, 3964646464164 = 19911422.
The squares which begin with 64 and end in 64 are
6461764 = 25422, 64128064 = 80082, 64673764 = 80422, 64931364 = 80582,
640494864 = 253082,...
64 is the unique 2-digit square with even digits.
1 / 64 = 0.015625, 15625 = 1252.
642± 3 are primes.
642 = 4096 is a zigzag square with distinct digits.
642 = (12 + 7)(32 + 7)(52 + 7) = (52 + 7)(112 + 7).
642 = 45 + 45 + 45 + 45.
642 = 4096, 4 + 0 + 96 = 102,
642 = 4096, 402 + 962 = 1042.
6461764 = 25422.
A cubic polynomial : (X + 642)(X + 1712)(X + 4322) = X3 + 4692X2 + 796322X +47278082.
Komachi equations:
642 = 98 * 7 * 6 + 5 - 4 * 3 * 2 - 1 = 98 * 7 * 6 - 5 - 4 * 3 - 2 - 1
= 98 * 7 * 6 - 5 * 4 + 3 - 2 - 1 = 98 * 7 * 6 - 5 * 4 - 3 + 2 + 1
= 98 * 7 * 6 - 54 / 3 - 2 * 1 = 98 * 7 * 6 - 54 / 3 - 2 / 1
= 9 * 8 * 7 * 65 * 4 / 32 + 1 = 9 / 8 * 7 * 65 / 4 * 32 + 1
= - 9 + 8 + 76 * 54 - 3 * 2 - 1 = 98 * 7 * 6 - 5 - 4 - 3 + 2 - 10
= 98 * 7 * 6 - 5 * 4 * 3 / 2 + 10 = 9 - 8 + 76 * 54 + 3 - 2 - 10
= 9 + 876 + 5 - 4 + 3210 = - 9 - 8 + 76 * 54 - 3 + 2 + 10,
642 = 12 + 22 - 32 * 42 - 52 * 62 - 72 + 82 * 92 = 12 - 22 + 32 * 42 + 52 * 62 + 72 * 82 - 92
= 12 * 22 - 32 - 42 + 52 * 62 + 72 * 82 + 92 = 122 * 32 / 42 + 562 / 72 * 82 - 92
= 12 * 22 - 342 + 562 / 72 + 82 * 92 = 12 + 232 * 42 - 52 - 672 + 82 + 92
= - 12 - 22 * 32 + 42 + 52 * 62 + 72 * 82 + 92 = - 122 * 32 / 42 + 562 / 72 * 82 + 92
= - 12 * 232 + 42 - 52 + 672 + 82 + 92 = - 12 * 22 * 32 - 452 - 62 * 72 + 892
= 92 + 82 * 72 + 62 * 52 + 42 - 32 * 22 - 12 = 92 + 82 * 72 + 62 * 52 - 42 - 32 + 22 */ 12
= 92 * 82 - 72 - 62 * 52 - 42 * 32 + 22 + 12 = 92 - 82 + 72 * 62 + 52 + 432 + 212
= - 92 + 82 * 72 + 62 * 52 + 42 * 32 - 22 + 12 = 92 - 82 - 72 + 652 + 42 - 32 - 22 - 102
= 92 * 82 + 72 * 62 - 542 - 32 * 22 + 102 = 92 + 82 * 72 - 62 - 52 + 42 + 322 - 102
= - 92 * 82 + 762 + 52 * 42 * 32 + 22 - 102,
642 = 123 * 33 * 43 * 563 / 73 / 83 / 93 = 123 * 33 * 43 / 563 * 73 * 83 / 93,
642 = 94 + 84 - 74 / 64 * 544 * 34 / 214 = - 94 + 84 + 74 / 64 * 544 * 34 / 214,
(642 + 8) = (72 + 8)(82 + 8), (642 - 8) = (82 - 8)(92 - 8).
642 + 652 + 662 + ... + 3052 = 30692.
(13 + 23 + ... + 93)(103 + 113 + ... + 153)(163 + 173 + ... + 643) = 103950002,
(13 + 23 + ... + 93)(103 + 113... + 393)(403 + 413 + ... + 643) = 675675002,
(13 + 23 + ... + 633)(643) = 10321922.
(12 + 22)(32 + 42)(52 + 62 + ... + 152)(162 + 172 + ... + 642) = 1155002,
(12 + 22 + ... + 62)(72)(82 + 92 + ... + 452)(462 + 472 + ... + 642) = 28442052,
(12 + 22 + ... + 192)(202 + 212 + ... + 282)(292 + 302 + ... + 322)(332 + 342 + ... + 642) = 613548002.
647 = 4398046511104, 4 + 3980 + 46 + 51 + 1 + 10 + 4 = 642,
647 = 4398046511104, 4 + 3980 + 46 + 51 + 11 + 04 = 642,
648 = 281474976710656, 2814 + 7 + 497 + 6 + 710 + 6 + 56 = 642,
649 = 18014398509481984, 1 + 8 + 014 + 3985 + 09 + 48 + 19 + 8 + 4 = 642,
1 + 801 + 439 + 850 + 9 + 4 + 8 + 1984 = 18 + 01 + 4 + 3985 + 09 + 48 + 19 + 8 + 4
= 18 + 014 + 3985 + 09 + 48 + 1 + 9 + 8 + 4 = 1801 + 439 + 850 + 9 + 4 + 8 + 1 + 984 = 642.
642 = 4096 appears in the decimal expressions of π and e:
π = 3.14159•••4096••• (from the 27372nd digit),
e = 2.71828•••4096••• (from the 5511st digit).
Page of Squares : First Upload February 2, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan