Take any four digits (four digits are exceptions to the same number), recombine the four digits that make up the number into the possible maximum number and the possible minimum number, and then find out the difference between them; Repeat the same process for this difference (for example, take 8028 at the beginning, the maximum recombination number is 8820, and the minimum recombination number is 0288, and the difference between them is 8532. Repeat the above process to get 8532-2358 = 6 174), and finally always reach the capra Karl black hole: 6 174. Calling it a "black hole" means that if you continue to operate, you will repeat this number and cannot "escape". The above calculation process is called capra Karl operation, and this phenomenon is called convergence. The result of 6 174 is called convergence result.

1. Any n digits will converge like 4 digits (1 and 2 digits are meaningless). 3 digits converge into a unique number 495; Four digits converge into a unique number 6174; 7 digits converge to a unique array (8 7-digit cyclic arrays _ _ _ are called convergence groups); There are several convergence results of other digits, including convergence numbers and convergence groups (for example, the convergence results of 14 digits _ * * and 9× 10 and 13 power _ _ _ have 6 convergence numbers and 2 1 convergence groups).

Once the convergence result is entered, the continuation of Caprai-Karl operation will be repeated in the convergence result, and it can no longer be "escaped".

Numbers in a convergent group can be exchanged in a progressive order (such as a → b → c or b → c → a or c → a → b).

Convergence results can be obtained without Caprai-Karl operation.

The number of convergence results of a given positioning number is finite and certain.

Second, the convergence result of a number with more digits (called n) is the convergence result of a number with less digits (called n, N﹥n), which is embedded in some specific numbers or arrays to form the convergence result of .4, 6, 8, 9, 1 1, 13.

The first is the number pair type, with two pairs:1) 9,02) 3,6.

The second type is the array type, which has one set:

7,2

5,4

1,8

The third category is numbers, there are two kinds:

1) 5 9 4

2) 8 6 4 2 9 7 5 3 1

2. Part of the embedding number is embedded in the back neighbor position greater than or equal to the last bit of the previous embedding number. The other part is embedded in the corresponding position in the back section _ _ _ _ _ _ _ _ _ _, and forms a layered group number structure with the number embedded in the front section.

594 can only be embedded with numbers like n=3+3k. Such as 9, 12, 15, 18. ...

The logarithm of 3, (9,0) (3,6) can be embedded alone or in combination with array type and number type.

rank

7,2

5,4

1,8

Must be "matched" and embedded in sequence: (7,2) → (5,4 )→ (1,8); Or (5,4) → (1,8 )→ (7,2)

Or (1, 8) → (7,2 )→ (5,4).

4. It can be embedded once, twice or multiple times (multiple digits will form a convergence result).

The convergence result of any N-digit is "hidden" in these N-digits, and Caprai-Carr operation only finds them out, not newly created.

Mention the phenomenon of "6 174 mathematical black hole"

1. American new scientist, 1992, 12, 19.

2. China reference news,1993,3, 14- 17.

3. Wang Jingzhi: (1) On the "black hole" in mathematics-on capra Karl constant.

⑵ Simplified some results of my calculus.

4. Tianshan Grass: A program that can perform arbitrary multi-bit capra-Lek operations.