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23

The smallest squares containing k 23's :
2304 = 482,   232324 = 4822,   2390623236 = 488942,
2391232342321 = 15463612,   14623123112323236 = 1209261062.

The first integer which is the sum of 8 squares in just two ways (see 20).

Every integer greater than 23 is the sum of 10 nonzero squares.

232 = 529, 5 + 2 + 9 = 42.

232 = 529 is a zigzag square.

A1 + B1, A2 + B2, A3 + B3 are squares for A = 184 and B = 345:
  1841 + 3451 = 232, 1842 + 3452 = 3912, 1843 + 3453 = 68772.
In general {A = 184m2, B = 345m2} has the same property (m = 1,2,...).
The second primitive solution is {A = 147916017521041, B = 184783370001360} for which
A1 + B1 = 182400492, A2 + B2 = 2366939840138412, A3 + B3= 30896092025335141400892.

232 = 13 + 23 + 23 + 83 = 14 + 24 + 44 + 44.

(232 + 3) = (42 + 3)(52 + 3) = (12 + 3)(22 + 3)(42 + 3),
(232 - 7) = (52 - 7)(62 - 7) = (32 - 7)(42 - 7)(62 - 7).

49k + 98k + 170k + 212k are squares for k = 1,2,3 (232, 2932, 39372).

Komachi Fractions : 232 = 385641/729, (23/315)2 = 4761/893025.

Komachi equations:
232 = 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 34 equations,
232 = 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 46 equations,
232 = 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 32 equations,
232 = 12 * 22 * 32 * 42 - 562 / 72 - 82 + 92,
232 = 13 + 23 + 33 - 43 - 53 - 63 - 73 + 83 + 93,
232 = 93 + 83 - 73 - 63 - 53 - 43 + 33 + 23 + 13.

(1 + 2)(3 + ... + 15)(16 + ... + 23) = 2342,
(1 + 2 + ... + 4)(5 + 6 + ... + 16)(17 + ... + 23) = 4202,
(1 + 2 + ... + 10)(11 + 12 + ... + 21)(22 + 23) = 6602,
(1 + 2 + ... + 11)(12)(13 + 14 + ... + 23) = 3962,
(1 + 2 + ... + 11)(12 + 13 + ... + 20)(21 + 22 + 23) = 7922.

13 + 23 + ... + 233 = (1 + 2 + ... + 23)2 = 2762,
(13 + 23 + ... + 63)(73 + 83 + ... + 203)(213 + 223 + 233) = 7858622.

232 = (14 + 22 + 33) + (44 + 52 + 63).

The sum of consecutive primes 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 102

None of the binomial coefficients nCk are square-free for every integer n > 23.

In the set {23, 1346, 4738, 8258} the sum of any two numbers is a square.

12 + 22 + 32 + ... + 232 = 4324, which consists of digits < 5.

232 = 529, 5 + 2 * 9 = 23.

23 is the first prime p for which the Legendre symbol (a/p) = 1 for a = 1,2,3,4.

232 = 529 appears in the decimal expressions of π and e:
  π = 3.14159•••529••• (from the 1058th digit),
(529 is the 9th 3-digit square in the expr. of π,)
  e = 2.71828•••529••• (from the 1265th digit).


Page of Squares : First Upload October 13, 2003 ; Last Revised January 25, 2011
by Yoshio Mimura, Kobe, Japan