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33

The smallest squares containing k 33's :
3364 = 582,   1336336 = 11562,   3333330225 = 577352,
1433333333961 = 11972192,   23383333333377801 = 1529160992.

Every integer greater than 33 can be represented by sums of 5 nonzero squares.

The 2nd integer which is the sum of 3 squares in just 2 ways: 12 + 42 + 42 = 22 + 22 + 52.
The 2nd integer which is the sum of 6 squares in just 3 ways (see 32).

332 is the second square which is the sum of 4 fifth powers : 15 + 25 + 25 + 45.

332 = 13 + 43 + 83 + 83.

332 = 43 + 45 + 17.

332 = (12 + 2)(192 + 2).

332 = 1089, its digits being distinct.

332 = 1089, which is reversible (9801 = 992). The unique reversible square of 4 digits.

Every integer greater than 33 can be the sum of 5 nonzero squares.

332± 2 are primes (the 5th case).

33k + 205k + 441k + 477k are squares for k = 1,2,3 (342, 6822, 142462).
113k + 124k + 262k + 590k are squares for k = 1,2,3 (332, 6672, 150572).

336 = 1291467969, 1 + 2 + 914 + 67 + 96 + 9 = 12 + 91 + 4 + 6 + 7 + 969 = 332,
337 = 42618442977, 4 + 2 + 618 + 442 + 9 + 7 + 7 = 332,
    42 + 6 + 18 + 4 + 42 + 977 = 42 + 6 + 18 + 44 + 2 + 977 = 332,
338 = 1406408618241, 1 + 40 + 64 + 0861 + 82 + 41 = 332,
    14 + 06 + 408 + 618 + 2 + 41 = 140 + 640 + 86 + 182 + 41 = 332,
339 = 46411484401953,
    4 + 64 + 1 + 14 + 8 + 4 + 40 + 1 + 953 = 4 + 64 + 1 + 14 + 8 + 44 + 01 + 953
  = 4 + 64 + 11 + 4 + 8 + 4 + 40 + 1 + 953 = 4 + 64 + 11 + 4 + 8 + 44 + 01 + 953
  = 4 + 64 + 11 + 48 + 4 + 4 + 01 + 953 = 4 + 64 + 114 + 844 + 01 + 9 + 53
  = 4 + 641 + 14 + 8 + 4 + 401 + 9 + 5 + 3 = 464 + 1 + 1 + 484 + 40 + 1 + 95 + 3
  = 464 + 114 + 8 + 4 + 401 + 95 + 3 = 464 + 114 + 8 + 440 + 1 + 9 + 53 = 332.

Komachi Fraction : (25/33)2 = 16875/29403.

Komachi equations:
332 = 1 + 23 * 45 + 6 + 7 * 8 - 9 = 1 * 234 * 5 + 6 - 78 - 9
  = 1 * 23 * 4 * 5 + 6 + 7 * 89,
332 = 9 * 8 + 765 * 4 / 3 - 2 - 1 = 9 + 8 - 7 - 6 + 543 * 2 - 1
  = 9 - 8 + 7 - 6 + 543 * 2 + 1, and more 10 equations,
332 = 9 + 876 * 5 / 4 - 3 - 2 - 10 = 9 + 876 * 5 / 4 - 3 / 2 * 10
  = 98 + 7 + 6 * 54 * 3 + 2 + 10, and more 14 equations,
332 = 92 * 82 - 72 - 62 - 52 - 42 - 32 * 212,
332 = 13 + 23 + 33 + 43 - 53 + 63 - 73 + 83 + 93 = 123 + 33 + 43 - 563 / 73 / 83 - 93,
332 = 93 + 83 - 73 + 63 - 53 + 43 + 33 + 23 + 13,
332 = 14 + 24 + 34 + 44 - 54 + 64 - 74 - 84 + 94,
332 = 94 - 84 - 74 + 64 - 54 + 44 + 34 + 24 + 14.

The 4-by-4 magic squares consisting of different squares with constant 332:

02 42172282
62302122 32
182132202142
272 22162102

(332 + 3) = (52 + 3)(62 + 3) = (12 + 3)(22 + 3)(62 + 3),
(332 - 9) = (62 - 9)(72 - 9).

(1 + 2 + ... + 32)(33) = 1322,
(1 + 2 + ... + 4)(5 + 6 + ... + 24)(25 + 26 + ... + 33) = 8702,
(1 + 2 + ... + 28)(29)(30 + 31 + ... + 33) = 12182.

13 + 23 + ... + 333 = (1 + 2 + ... + 33)2 = 5612

(12)(22 + 32 + ... + 102)(112 + 122 + ... + 172)(182 + 192 + ... + 352)= 840002,
(12 + 22 + ... + 52)(62 + 72 + ... + 102)(112 + 122 + ... + 172)(182 + 192 + ... + 352)= 5775002,
(12 + 22 + ... + 92)(102 + 112 + ... + 172)(182 + 192 + ... + 272)(282 + 292 + ... + 352)= 41895002,
(12 + 22 + ... + 102)(112)(122 + 132 + ... + 242)(252 + 262 + ... + 352) = 14314302.

332 = 1089 appears in the decimal expressions of π and e:
  π = 3.14159•••1089••• (from the 2534th digit),
  e = 2.71828•••1089••• (from the 10421st digit).


Page of Squares : First Upload November 17, 2003 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan