The smallest squares containing k 32's :
324 = 182, 232324 = 4822, 3232832164 = 568582,
563223232324 = 7504822, 323232323267569 = 179786632.
The second integer which is the sum of a square and a prime in just 3 ways :
12 + 31, 32 + 23, 52 + 7.
The first integer which is the sum of 8 squares in just 4 ways:
[1, 1, 1, 1, 1, 1, 1, 5], [1, 1, 1, 1, 1, 3, 3, 3], [1, 1, 1, 1, 2, 2, 2, 4], [2, 2, 2, 2, 2, 2, 2, 2]
The 2nd integer which is the sum of 5 squares in just 3 ways:
[1, 1, 1, 2, 5], [1, 2, 3, 3, 4], [2, 2, 2, 2, 4].
322 = 83 + 83 = 23 + 23 + 23 + 103 = 44 + 44 + 44 + 44 = 28 + 28 + 28 + 28 = 29 + 29.
322 = 1024, its digits being distinct.
322 = (12 + 7)(112 + 7).
322 = 1024, 1 + 0 + 24 = 52,
322 = 1024, 10 + 2 + 4 = 42,
322 = 1024, 102 + 242 = 262.
326 = 1073741824, 12 + 072 + 32 + 72 + 42 + 182 + 242 = 322.
328 = 1099511627776,
1 + 09 + 95 + 116 + 27 + 776 = 1 + 09 + 951 + 16 + 27 + 7 + 7 + 6
= 1 + 099 + 5 + 116 + 27 + 776 = 10 + 9 + 9 + 51 + 162 + 7 + 776
= 10 + 9 + 9 + 51 + 162 + 777 + 6 = 10 + 9 + 951 + 1 + 6 + 27 + 7 + 7 + 6
= 109 + 9 + 5 + 116 + 2 + 7 + 776 = 109 + 9 + 5 + 116 + 2 + 777 + 6
= 109 + 95 + 1 + 16 + 27 + 776 = 109 + 95 + 11 + 6 + 27 + 776 = 322.
Cubic Polynomial (X + 322)(X + 22052)(X + 42242) = X3 + 47652X2 + 93151682X + 2980454402
Komachi equations:
322 = 12 + 34 * 5 * 6 - 7 + 8 - 9 = 1 + 23 * 45 - 6 - 7 - 8 + 9
= 12 * 3 * 4 * 56 / 7 * 8 / 9, and more 4 equations,
322 = 987 + 6 + 5 * 4 * 3 / 2 + 1 = 987 + 6 * 5 + 4 * 3 / 2 + 1
= 987 + 6 * 5 * 4 / 3 - 2 - 1, and more 15 equations,
322 = 9 * 87 + 6 - 5 + 4 * 3 * 2 * 10 = 987 + 6 + 5 * 4 + 3 - 2 + 10
= 987 + 6 * 5 - 4 + 3 - 2 + 10, and more 11 equations,
322 = 12 / 22 * 342 * 52 - 62 - 782 - 92 = 1232 / 42 + 52 - 62 * 72 / 82 + 92,
322 = 92 + 82 - 72 + 62 * 52 + 42 + 32 + 22 - 12,
322 = 123 * 33 / 43 + 563 / 73 + 83 - 93 = - 123 * 33 / 43 + 563 / 73 + 83 + 93
= 13 + 23 / 33 * 453 / 63 - 73 + 83 + 93 = 13 + 23 / 33 / 43 * 53 * 63 - 73 + 83 + 93,
322 = 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,
322 = 93 + 83 + 73 + 63 + 53 + 43 + 33 + 23 - 103,
322 = 124 * 34 / 44 + 54 - 64 - 74 + 84 - 94 = - 124 * 34 / 44 + 54 - 64 - 74 + 84 + 94.
(322 + 2) = (22 + 2)(132 + 2) = (52 + 2)(62 + 2).
92 + 102 + 112 + ... + 322 = 1062.
322 + 332 + 342 + ... + 40872 = 1508782,
322 + 332 + 342 + ... + 6092 = 86872,
322 + 332 + 342 + ... + 612812 = 87585752,
322 + 332 + 342 + ... + 1488562 = 331582102.
(1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + ... + 32) = 3002,
(1 + 2 + 3 + 4)(5 + 6 + ... + 19)(20 + 21 + ... + 32) = 7802,
(1 + 2 + ... + 9)(10 + 11 + ... + 17)(18 + 19 + ... + 32) = 13502.
(12 + 22 + 32)(42 + 52 + 62 + 72)(82)(92 + 102 +... + 322)= 356162.
(12 + 22 + 32 + ... + 322) = 11440, which consists of 3 kinds of digits.
13 + 23 + ... + 323 = (1 + 2 + ... + 32)2 = 5282.
322 = 1024 appears in the decimal expressions of π and e:
π = 3.14159•••1024••• (from the 12735th digit),
e = 2.71828•••1024••• (from the 3109th digit).
Page of Squares : First Upload November 17, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan