The 3rd integer which is the sum of 2 squares: 22 + 22.
The 2nd integer which is the sum of 5 squares.
The smallest squares containing k 8's are :
81 = 92, 6889 = 832, 88804 = 2982, 1898884 = 13782,
78818884 = 88782, 878588881 = 296412, 88888055881 = 2981412,
388888184881 = 6236092, 86888087818884 = 93213782.
82 = 64, the first square with a decreasing sequence of digits.
8 and 9 are consecutive integers having square factors (the first case).
82± 3 are primes.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 62,
(1 + 2) x (3) x (4 + 5 + 6) x (7 + 8) = 452, (1) x (2) x (3 + 4 + 5 + 6 + 7) x (8) = 202,
(1) x (2 + 3 + 4) x (5 + 6 + 7) x (8) = 362, (1 + 2) x (3) x (4) x (5 + 6 + 7) x (8) = 722.
13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) = (62)2,
(13) x (23 + 33) x (43 + 53 + 63 + 73 + 83) = 2102,
(13 + 23 + 33) x (43 + 53) x (63) x (73) x (83) = 5080322
1 + 25 + 3 + 45 + 5 + 65 + 7 + 85 = 2042.
12 + 22 + ... + 82 = 204, every digit of which is even.
The next such case occurs for 12 + 22 + ... + 203432.
82 is the first square which is the sum of 5 cubes: 13 + 13 + 23 + 33 33.
82 is the third square which is the sum of 4 fourth powers: 24 + 24 + 24 + 24.
82 is the first square which is the sum of 2 fifth powers: 25 + 25.
82 - 72 + 62 - 52 + 42 - 32 + 22 - 12 = 62.
82 = 64 is the first square which consists of even digits.
82 + 2 = (22 + 2)(32 + 2), 82 - 4 = (32 - 4)(42 - 4),
82 + 6 = (12 + 6)(22 + 6).
Komachi Fractions: 82 = 372096/5814
(8/11)2 = 19584/37026 = 39168/74052, (8/31)2 = 8640/129735, (8/53)2 = 8640/379215
Komachi equations:
82 = 1 + 2 + 3 + 4 + 5 * 6 + 7 + 8 + 9 = 1 + 2 + 3 + 4 - 5 + 6 * 7 + 8 + 9
= 1 + 2 + 3 + 4 - 5 - 6 + 7 * 8 + 9, and more 238 equations,
82 = 9 + 8 + 7 + 6 * 5 + 4 + 3 + 2 + 1 = 9 + 8 + 7 + 6 * 5 + 4 + 3 * 2 * 1
= 9 + 8 + 7 + 6 * 5 + 4 * 3 - 2 * 1,, and more 262 equations
82 = 9 + 8 + 7 * 6 + 5 + 4 + 3 * 2 - 10 = 9 + 8 + 7 * 6 + 5 + 4 * 3 - 2 - 10
= 9 + 8 + 7 * 6 + 5 - 4 - 3 * 2 + 10, and more 174 equations,
82 = 122 - 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, and more 9 equations,
82 = 92 + 82 - 72 / 62 * 542 * 32 / 212 = 92 + 82 - 72 - 62 + 52 + 42 - 32 * 22 - 12
= 92 + 82 - 72 - 62 + 52 - 42 - 32 + 22 * 12, and more 15 equations,
82 = 982 / 72 - 62 + 52 - 42 - 32 + 22 - 102 = - 982 / 72 / 62 + 52 + 42 / 32 / 22 * 102
= 92 + 82 - 72 - 62 / 52 * 42 - 32 + 22 / 102, and more 3 equations,
82 = 13 + 23 * 33 + 43 + 53 - 63 - 73 - 83 + 93 = 13 - 23 * 33 + 43 + 53 + 63 - 73 - 83 + 93
= 13 * 23 + 33 + 43 - 53 + 63 - 73 - 83 + 93, and more 3 equations,
82 = 93 - 83 - 73 + 63 + 53 + 43 - 33 * 23 + 13 = 93 - 83 - 73 + 63 - 53 + 43 + 33 + 23 * 13
= 93 - 83 - 73 - 63 + 53 + 43 + 33 * 23 + 13, and more 5 equations,
82 = 96 * 86 / 76 / 66 / 546 * 36 * 216 = 986 / 76 / 66 * 546 / 36 / 216.
In the set {8, 953, 4088, 8456} the sum of any two numbers is a square :
8 + 953 = 312, 8 + 4088 = 642, 8 + 8456 = 922,
953 + 4088 = 712, 953 + 8456 = 972, 4088 + 8456 = 1122.
Also the set {8, 1016, 1288, 3473} has the same property.
88 = 16777216, 16 + 7 + 7 + 7 + 21 + 6 = 82,
810 = 1073741824, 1 + 07 + 3 + 7 + 4 + 18 + 24 = 1 + 07 + 37 + 4 + 1 + 8 + 2 + 4
= 10 + 7 + 3 + 7 + 4 + 1 + 8 + 24 = 82,
812 = 68719476736, 6 + 8 + 7 + 1 + 9 + 4 + 7 + 6 + 7 + 3 + 6 = 82,
814 = 4398046511104, 4 + 3 + 9 + 8 + 04 + 6 + 5 + 11 + 10 + 4 = 82.
82 = 64 appears in the decimal expressions of π and e:
π = 3.14159•••64••• (from the 22nd digit), 64 is the first 2-digit square in the expr. of π,
e = 2.71828•••64••• (from the 96th digit), 64 is the 6th 2-digit square in the expr. of e.
Page of Squares : First Upload August 24, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan