The smallest squares containingk 84's :
484 = 222, 228484 = 4782, 168428484 = 129782,
849847984384 = 9218722, 848447840784384 = 291281282.
The squares which begin with 84 and end in 84 are
84125584 = 91722, 84235684 = 91782, 842276484 = 290222, 842624784 = 290282,
845181184 = 290722,...
842 = 7056, a square with different digits.
842 = (1)(2 + 3 + ... + 5)(6)(7 + 8 + ... + 14).
842 = 7056, 70 + 5 + 6 = 92.
842 = 63 + 143 + 163.
842 = 7056, 70 / 5 * 6 = 84.
842 = (12 + 3)(32 + 3)(122 + 3) = (22 + 3)(32 + 3)(92 + 3).
842 = (13 + 1)(33 + 1)(53 + 1).
18k + 84k + 129k + 210k are squares for k = 1,2,3 (212, 2612, 34652).
The integral triangle of sides 41, 357, 370 (or 28, 1345, 1371) has area 842, a square.
Komachi Fraction : (84/89)2 = 63504/71289.
Komachi equations:
842 = 12 / 3 * 4 * 56 * 7 / 8 * 9 = - 12 + 34 * 5 * 6 * 7 - 8 * 9
= - 9 * 8 * 7 + 6 * 5 * 4 * 3 * 21 = 9 * 87 / 6 * 54 - 3 + 2 + 10,
842 = 12 * 22 / 32 / 42 * 5672 * 82 / 92 = - 12 + 232 + 42 * 52 * 62 + 72 - 892
= - 12 + 22 / 32 * 452 - 62 * 72 + 892 = 92 * 82 * 72 / 62 * 52 / 42 - 32 * 212
= 982 / 72 * 62 * 52 / 42 - 32 * 212 = - 92 + 872 - 62 - 52 * 42 + 32 - 22 - 12
= - 92 + 82 + 762 + 542 / 32 * 22 + 12 = - 92 - 82 + 762 + 52 * 42 + 322 + 12.
84125584 = 91722 and 84235684 = 91782 are squares.
A cubic Polynomial :
(X + 842)(X + 8192)(X + 11682) = X3 + 14292X2 + 9640682X + 803537282.
(842 - 6) = (92 - 6)(102 - 6).
12 + 22 + 32 + 42 + ... + 842 = 201110, which consists of 3 kinds of digits.
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 84) = 2732,
(1 + 2 + ... + 20)(21 + 22 + ... + 84) = 8402.
842 = 25 x 26 + 27 x 28 + 29 x 30 + 31 x 32 + ... + 37 x 38.
842 = (2 + 3 + 4 + 5)2 + (6 + 7 + 8 + 9)2 + (10 + 11 + 12 + 13)2 + (14 + 15 + 16 + 17)2.
(12 + 22 + ... + 62)(72 + 82 + ... + 842) = 42772,
(12 + 22 + ... + 122)(132 + 142 + ... + 252)(262 + 272 + ... + 342)(352 + 362 + ... + 842) = 696150002,
(12 + 22 + ... + 162)(172 + 182 + ... + 392)(402 + 412 + ... + 512)(522 + 532 + ... + 842) = 3327941762.
847 = 29509034655744, 2 + 950 + 9 + 0346 + 5 + 5744 = 842,
848 = 2478758911082496, 2 + 4 + 7 + 8 + 7 + 5891 + 1082 + 49 + 6 = 842,
848 = 2478758911082496, 2 + 47 + 8 + 7 + 5891 + 1082 + 4 + 9 + 6 = 842,
849 = 208215748530929664,
2+08+2+1+5748+530+92+9+664 = 2+08+2+15+748+5309+2+966+4
= 2+08+2157+4853+09+2+9+6+6+4 = 2+082+1574+8+5309+2+9+6+64
= 2+082+1574+8+5309+2+9+66+4 = 208+2+1+5748+5+30+92+966+4
= 2082+1+5+7+4853+09+29+6+64 = 2082+1+5+7+4853+09+29+66+4 = 842.
842 = 7056 appears in the decimal expressions of π and e:
π = 3.14159•••7056••• (from the 7119th digit),
e = 2.71828•••7056••• (from the 18244th digit).
Page of Squares : First Upload March 1, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan