所有 999...999 的因數,除了3 之外,以因數為分母的所有真分數的循環小數,"只有其中一個循環節的平方,分半和等於原循環節"成立.

1/3 = 0.33333...
2/3 = 0.66666...

3^2 = 9 =/= 3

6^2 = 36 -> 3+6 = 9 =/= 6

KM 為 M/Q 的循環節
QN 為 N/K 的循環節

任何四個正整數,K,Q,M,N

滿足下面兩個式子,

KxQ = 999...999( N個9)
KM + QN = 1000...000( N個0)

則 (KM)^2 為Kaprekar number

1/17
循環節0588235294117647

0588235294117647^ 2 = 346020761245674571280276816609
->4705882352941176

1/7 = abcdef
(abcdef -1)^2 = Xx = X + x = ABCDEF

142857 - 1 ->857143 -> 857142 + 1
428571 - 1 -> 428571 + 1
285714 - 1 -> 1
857142 - 1 -> 428571 + 1
571428 -1 -> 142857 + 1
714285 - 1 -> 142857 + 1

1/13 = abcdef
2/13 = ghijkl
(abcdef + 1)^2 = Xx = X + x
(ghijkl + 1)^2 = Xx = X + x

076924 -> 307692 + 1
153847 -> 923076 + 1
230770 -> 846153 + 1
307693 -> 76923 + 1
384616 -> 615384 + 1
461539 -> 461539
538462 -> 615384 + 1
615385 -> 76923 + 1
692308 -> 846153 + 1
769231 -> 923076 + 1
846154 -> 307692 + 1
923077 -> 1

1/7 = abcdef
(abcdef +1)^2 = Xx = X + x = ABCDEF

142858 ->428572 -> 428571 + 1
428572 -> 142858 -> 142857 + 1
285715 -> 142858 -> 142857 + 1
857143 -> 857143
571429 -> 428572 -> 428571 + 1
714286 -> 1

1/19 = 052631578947368421...

736842105263157894 & 263157894736842106

263157894736842106
/19 = 13850415512465374

1/17 = 0588235294117647 ...

8823529411764705
& 1176470588235295

8823529411764705 ^ 2 = AB -> A+B = 8823529411764705
& 1176470588235295^ 2 = AB -> A+B =
1176470588235295


1176470588235295/17 = 69204152249135

1/13 = abcdef
2/13 = ghijkl
(abcdef)^2 = Xx = X + x = ghijkl
(ghijkl)^2 = Xx = X + x = ghijkl

076923 -> 153846 -> 615384 -> 846153 -> 615384 ...
153846 -> 615384-> 846153
-> 615384 ...

230769 -> 384615 -> 846153 -> 615384 ...
307692 -> 461538 -> 538461
384615 -> 846153 -> 615384 -> 846153 -> 615384 ...

461538 -> 538461
538461 -> 538461
615384 -> 846153 -> 615384 -> 846153 -> 615384 ...

692307 -> 461538 -> 538461
769230 -> 384615 -> 846153 -> 615384 ...

846153 -> 615384 ...
923076 -> 153846 -> 615384 -> 846153 -> 615384

1/17
循環節052531578947368421

052531578947368421^ 2 = xxxxx...xxxx
-> 789473684210525315

857142 ->-> 142857

857143 -> 857143 (!)

41 X 271 = (99999) /9 = 11111

1/41 = abcde
1/271 = ABCDE
(abcde)^2 = Xx = X + x = fghij
(ABCDE)^2 = Yy = y + Y = FGHIJ

1/7 = abcdef
(abcdef)^2 = Xx = X + x = ABCDEF

142857 -> 142857
428571 -> 285714
285714 -> 571428
857142 ->-> 142857
571428 ->-> 285714
714285 -> 571428

1/13 = abcdef
2/13 = ghijkl
(abcdef)^2 = Xx = X + x = ghijkl
(ghijkl)^2 = Xx = X + x = ghijkl

076923 -> 153846
153846 -> 615384
230769 -> 384615
307692 -> 461538
384615 -> 846153
461538 -> 538461
538461 -> 538461
615384 -> 846153
692307 -> 461538
769230 -> 384615
846153 -> 615384
923076 -> 153846