The smallest squares containing k 51's :
5184 = 722, 51051025 = 71452, 35151375169 = 1874872,
15165151908516 = 38942462, 51251185151464516 = 2263872462.
The 5th integer which is the sum of 4 distinct squares: 12 + 22 + 32 + 62.
The third integer which is the sum of 3 squares in just 2 ways (see 50).
The first integer which is the sum of 13 squares in just 6 ways (see 49).
1 / 51 = 0.0196..., 196 = 142.
512 = 2601 is a zigzag square whose digits are distinct.
302 + 512 = 3501, 702 + 512 = 7501.
512 = 2601, 2 + 6 + 0 + 1 = 32, 23 + 63 + 03 + 13 = 152.
516 = 17596287801, 1759 + 6 + 28 + 7 + 801 = 512,
519 = 2334165173090451,
2 + 3 + 341 + 65 + 1730 + 9 + 0451 = 2 + 334 + 1651 + 73 + 090 + 451 = 512,
23 + 3 + 4 + 1651 + 7 + 3 + 0904 + 5 + 1 = 2334 + 1 + 6 + 51 + 73 + 090 + 45 + 1 = 512,
2334 + 165 + 1 + 7 + 30 + 9 + 04 + 51 = 2334 + 165 + 17 + 30 + 9 + 045 + 1 = 512.
18k + 282k + 921k + 1380k are squares for k = 1,2,3 (512, 16832, 585812).
51k + 1683k + 6069k + 10693k are squares for k = 1,2,3 (1362, 124102, 12045522).
Komachi equations:
512 = 9 + 8 * 7 * 6 * 54 * 3 / 21 = 9 + 8 / 7 * 6 * 54 / 3 * 21
= - 9 + 87 * 6 * 5 - 4 + 3 + 2 - 1 = - 9 + 87 * 6 * 5 + 4 - 3 - 2 + 1
= 9 * 87 / 6 * 5 * 4 + 3 - 2 - 10 = 9 + 87 * 6 * 5 + 4 - 32 + 10
= - 9 + 87 * 6 * 5 - 4 - 3 * 2 + 10 = - 9 + 87 * 6 * 5 - 4 * 3 + 2 + 10
= - 9 + 87 * 6 * 5 + 4 + 3 * 2 - 10 = - 9 + 87 * 6 * 5 + 4 * 3 - 2 - 10
= - 9 * 8 - 7 * 6 + 543 / 2 * 10,
512 = 122 - 32 - 42 + 562 * 72 / 82 + 92 = - 92 - 82 + 72 * 62 - 52 - 42 + 322 - 12
= - 92 + 82 * 72 - 62 - 52 + 42 - 32 - 22 * 102.
(512 - 5) = (72 - 5)(82 - 5) = (32 - 5)(42 - 5)(82 - 5),
(512 + 9) = (62 + 9)(72 + 9).
(1 + 2)(3 + 4 + ... + 51)=632,
(1 + 2 + 3 + 4)(5 + 6 + ... + 11)(12 + 13 + ... + 51)=8402,
(1 + 2 + ... + 9)(10 + 11 + ... + 44)(45 + 46 + ... + 51)=37802,
(1 + 2 + ... + 12)(13 + 14 + ... + 51)=3122,
(1 + 2 + ... + 26)(27 + 28 + ... + 51)=5852,
(1 + 2 + ... + 32)(33 + 34 + ... + 47)(48 + 49 + 50 + 51)=79202,
(1 + 2 + ... + 48)(49 + 50 + 51)=4202,
(1 + 2 + ... + 50)(51)=2552.
(12 + 22 + ... + 72)(82 + 92 + ... + 152)(162 + 172 + ... + 372)(382 + 392 + ... + 512)=83853002.
512 = 2601 appears in the decimal expressions of π and e:
π = 3.14159•••2601••• (from the 2064th digit),
e = 2.71828•••2601••• (from the 3008th digit).
Page of Squares : First Upload January 19, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan