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10

The 4th number which is the sum of 2 squares: 12 + 32.

The smallest squares containing k 10's :
100 = 102,   1010025 = 10052,   1010604100 = 317902,
108310101025 = 3291052,   1010091010961041 = 317819292.

10 = (12 + 22 + 32 + 42 + 52 + 62 + 72) / (12 + 22 + 32).

102 is the first square which is the sum of 5 distinct squares: 12 + 32 + 42 + 52 + 72.
102 is the first square which is the sum of 4 squares in just 5 ways:
12 + 12 + 72 + 72 = 12 + 32 + 32 + 92 = 12 + 52 + 52 + 72 = 22 + 42 + 42 + 82 = 52 + 52 + 52 + 52.
102 is the first square which is the sum of 4 distinct cubes: 13 + 23 + 33 + 43.

102± 3 are primes.

9k + 10k + 60k + 90k are squares for k = 1,2,3 (132, 1092, 9732).
10k + 44k + 57k + 58k are squares for k = 1,2,3 (132, 932, 6832).
10k + 74k + 98k + 142k are squares for k = 1,2,3 (182, 1882, 20522).
10k + 218k + 542k + 674k are squares for k = 1,2,3 (382, 8922, 218122).
10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).

10k + 50k + 190k + 320k + 330k are squares for k = 1,2,3,4 (302, 5002, 87002, 1538002).

102 + 112 + 122 = 132 + 142.

102 = (1 + 2 + 3 + 4)2 = 13 + 23 + 33 + 43.

102 = 1 x 2 + 3 x 4 + 5 x 6 + 7 x 8.

102 = (12 + 1)(22 + 1)(32 + 1) = (12 + 1)(72 + 1) = (12 + 4)(42 + 4).

102 = 26 + 62.

102 = (1) x (2 + 3) x (4) x (5) = (12) x (22) x (32 + 42).

102 is the sum of consecutive primes : 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23.

(102 - 2) = (32 - 2)(42 - 2),   (102 + 4) = (22 + 4)(32 + 4),
(102 + 8) = (12 + 8)(22 + 8).

Komachi equations:
102 = 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 159 equations,
102 = 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 213 equations,
102 = 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 140 equations,
102 = 12 * 2342 * 52 * 62 / 782 / 92 = - 122 / 32 + 452 - 62 * 72 - 82 - 92
  = 12 + 232 - 42 * 52 + 62 - 72 + 82 - 92, and more 14 equations,
102 = 982 / 72 - 62 - 52 - 42 * 32 / 22 + 12 = 92 + 82 - 72 + 62 + 52 * 42 + 32 - 212
  = - 92 + 82 + 72 + 62 - 52 * 42 - 32 + 212 = - 982 + 762 - 52 - 42 + 32 * 212,
102 = 92 + 82 * 72 - 652 - 42 + 322 + 102 = - 92 - 82 * 72 + 652 + 42 - 322 + 102
102 = 92 + 82 - 72 + 62 / 52 / 42 / 32 * 22 * 102, and more 7 equations.

(1 + 2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10) = 212,
(1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + 10) = 272,   (1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10) = 302,
(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10) = 452,   (1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10) = 632,
(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10) = 702,   (1)(2 + 3 + 4 + 5)(6)(7)(8 + 9 + 10) = 1262,
(1 + 2 + 3)(4)(5 + 6 + 7)(8 + 9 + 10) = (1)(2)(3)(4)(5 + 6 + 7)(8 + 9 + 10) = 1082,
(1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9)(10) = 1802,   (1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9)(10) = 2702,
(1 + 2 + 3 + 4)(5 + 6 + 7)(8)(9)(10) = 3602,   (1 + 2)(3 + 4 + 5)(6)(7 + 8)(9)(10) = 5402,
(1 + 2)(3)(4 + 5)(6)(7 + 8)(9)(10) = 8102,   (1)(2 + 3)(4)(5)(6)(7 + 8)(9)(10) = 9002.

(12 + 22)(32 + 42)(52)(62)(72)(82 + 92 + 102) = 367502,
(12 + 22)(32)(42)(52)(62)(72)(82 + 92 + 102) = 882002.

13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 = (1 + 2 + ... + 10)2 = 552,
(13 + 23)(33)(43 + 53)(63 + 73 + 83 + 93 + 103) = 113402,
(13)(23 + 33 + 43 + 53 + 63)(73 + 83 + 93)(103) = 264002,
(13)(23 + 33)(43)(53)(63 + 73 + 83 + 93 + 103) = 280002,
(13 + 23 + 33 + 43)(53)(63 + 73 + 83 + 93)(103) = 1500002,
(13)(23)(33 + 43)(53 + 63 + 73)(83)(93 + 103) = 6639362,
(13 + 23 + 33)(43)(53)(63 + 73 + 83 + 93)(103) = 7200002,

102 = 100 appears in the decimal expression of e:
  e = 2.71828•••100••• (from the 1029th digit).


Page of Squares : First Upload September 1, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan