The square of 3.
The 3rd integer which is the sum of 3 squares: 12 + 22 + 22.
The 2nd integer which is the sum of 6 squares.
The smallest squares containing k 9's are :
9 = 32, 3969 = 632, 29929 = 1732, 1999396 = 14142,
299739969 = 173132, 2909199969 = 539372, 19299599929 = 1389232,
909995799969 = 9539372, 9499999990849 = 30822072.
The squares which begin with 9 and end in 9 are
9409 = 972, 91809 = 3032, 94249 = 3072, 97969 = 3132, 908209 = 9532,...
9 = (12 + 22 + 32 + ... + 132) / (12 + 22 + 32 + 42 + 52 + 62).
92 = 81, 8 + 1 = 32.
92 = 81, the second square with a decreasing sequence of digits.
92± 2 are primes (the second case).
92 is the first square which is the sum of 4 distinct squares: 22 + 42 + 52 + 62.
92 is the second square which is the sum of 3 distinct squares: 12 + 42 + 82.
92 is the first square which is the sum of 3 squares in just 3 ways:
12 + 42 + 82 = 32 + 62 + 62 = 42 + 42 + 72.
92 is the third square which is the sum of 4 squares in just 2 ways:
22 + 22 + 32 + 82 = 22 + 42 + 52 + 62.
92 is the second square which is the sum of 3 cubes: 33 + 33 + 33.
92 is the third square which is the sum of 4 cubes: 13 + 23 + 23 + 43.
92 = (1 + 2) x (3) x (4 + 5).
92 = (12 + 2)(52 + 2).
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) x (9) = 182, (1 + 2) x (3 + 4 + 5 + 6) x (7 + 8 + 9) = 362,
(1 + 2 + 3) x (4) x (5) x (6 + 7 + 8 + 9) = (1) x (2) x (3) x (4) x (5) x (6 + 7 + 8 + 9) = 602,
(1) x (2) x (3 + 4 + 5 + 6 + 7) x (8) x (9) = 602,
(1 + 2) x (3 + 4 + 5) x (6) x (7 + 8 + 9) = 722,
(1 + 2) x (3) x (4 + 5) x (6) x (7 + 8 + 9) = (1) x (2 + 3 + 4) x (5 + 6 + 7) x (8) x (9) = 1082,
(1) x (2 + 3) x (4) x (5) x (6) x (7 + 8 + 9) = 1202,
(1 + 2) x (3) x (4 + 5 + 6) x (7 + 8) x (9) = 1352,
(1 + 2) x (3) x (4) x (5 + 6 + 7) x (8) x (9) = 2162.
13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2.
92 + 102 + 112 + ... + 322 = 1062,
92 + 102 + 112 + ... + 5520572 = 2368186192.
(92 + 3) = (22 + 3)(32 + 3), (92 + 7) = (12 + 7)(22 + 7),
(92 - 3) = (32 - 3)(42 - 3), (92 + 3) = (22 - 3)(32 - 3)(42 - 3).
The first nontrivial Kaprekar number: 92 = 81, 8 + 1 = 9
The sum of the divisors of 92 is 112 (The next example is 202).
9 = 32 = 1! + 2! + 3!
13 + 33 + 53 + 73 + 93 = 352
(12 + 23 + 34) + (42 + 53 + 64) + (72 + 83 + 94) = 932
(13 + 23 + 35) + (43 + 53 + 65) + (73 + 83 + 95) = 2612
92 = 53 - 43 + 33 - 23 + 13.
92 + 102 + 112 + 122 + ... + 322 = 1062
9k + 10k + 60k + 90k are squares for k = 0,1,2,3 (22, 132, 1092, 9732).
Cubic polynomial (X + 92)(X + 282)(X + 4322) = X3 + 4332X2 + 127082X + 1088642.
92 = 40 + 42 + 43.
Komachi Fractions: (9/62)2 = 3645/172980, (9/733)2 = 729/4835601
Komachi equations:
92 = 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 200 equations,
92 = 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 260 equations,
92 = 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 168 equations,
92 = 122 - 32 * 42 + 562 - 72 * 82 + 92 = 12 - 22 * 32 + 452 - 62 * 72 - 82 - 92
= 12 - 22 - 32 + 42 - 52 + 62 + 72 - 82 + 92, and more 39 equations,
92 = 92 + 82 + 72 * 62 + 52 - 432 - 22 * 12 = 92 + 82 + 72 - 62 - 52 - 42 - 32 * 22 * 12
= 92 - 82 + 72 + 62 - 52 + 42 - 32 - 22 + 12, and more 11 equations,
92 = 92 + 82 - 72 * 62 + 52 / 42 * 322 + 102 = 92 - 82 + 72 * 62 - 52 / 42 * 322 - 102
= - 92 + 82 - 72 + 62 - 52 + 42 * 32 / 22 + 102 = - 982 / 72 - 62 + 52 * 42 + 32 + 22 - 102,
92 = 13 + 23 - 33 + 43 + 53 - 63 + 73 + 83 - 93,
92 = - 93 + 83 + 73 - 63 + 53 + 43 - 33 + 23 + 13.
93 = 729, 72 + 9 = 92,
96 = 531441, 5 + 31 + 4 + 41 = 5 + 31 + 44 + 1 = 92,
97 = 4782969, 47 + 8 + 2 + 9 + 6 + 9 = 92,
98 = 43046721, 4 + 3 + 04 + 67 + 2 + 1 = 4 + 3 + 046 + 7 + 21 = 43 + 04 + 6 + 7 + 21 = 92,
99 = 387420489, 3 + 8 + 7 + 4 + 2 + 048 + 9 = 3 + 8 + 7 + 42 + 04 + 8 + 9 = 92.
910 = 3486784401, 3 + 4 + 8 + 6 + 7 + 8 + 4 + 40 + 1 = 3 + 4 + 8 + 6 + 7 + 8 + 44 + 01
= 3 + 48 + 6 + 7 + 8 + 4 + 4 + 01 = 92,
911 = 31381059609, 3 + 1 + 38 + 10 + 5 + 9 + 6 + 09 = 31 + 3 + 8 + 10 + 5 + 9 + 6 + 09 = 92,
912 = 282429536481, 2 + 8 + 2 + 4 + 2 + 9 + 5 + 36 + 4 + 8 + 1 = 92,
913 = 2541865828329, 2 + 5 + 4 + 1 + 8 + 6 + 5 + 8 + 2 + 8 + 3 + 29
= 2 + 5 + 4 + 1 + 8 + 6 + 5 + 8 + 28 + 3 + 2 + 9
= 25 + 4 + 1 + 8 + 6 + 5 + 8 + 2 + 8 + 3 + 2 + 9 = 92,
915 = 205891132094649, 2 + 05 + 8 + 9 + 1 + 1 + 3 + 20 + 9 + 4 + 6 + 4 + 9
= 20 + 5 + 8 + 9 + 1 + 1 + 3 + 2 + 09 + 4 + 6 + 4 + 9 = 92,
915 = 205891132094649, 2052 + 892 + 112 + 32 + 22 + 0942 + 62 + 42 + 92 = 95,
916 = 1853020188851841, 1 + 8 + 5 + 3 + 02 + 018 + 8 + 8 + 5 + 18 + 4 + 1
= 1 + 8 + 5 + 3 + 020 + 1 + 8 + 8 + 8 + 5 + 1 + 8 + 4 + 1
= 18 + 5 + 3 + 02 + 01 + 8 + 8 + 8 + 5 + 18 + 4 + 1
= 18 + 5 + 3 + 02 + 018 + 8 + 8 + 5 + 1 + 8 + 4 + 1 = 92.
92 = 81 appears in the decimal expressions of π and e:
π = 3.14159•••81••• (from the 67th digit), 81 is the 4th 2-digit square in the expression of π,
e = 2.718281828••• (from the 5th digit), 81 is the first 2-digit square in the expression of e.
Page of Squares : First Upload August 24, 2003 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan