What kind of natural number n has the number of flowers? Is such n finite or infinite? For a given n, how many flowers, if any? At first, we thought that any n flowers had flowers, but in 1986, Anthony Diluna, an American math teacher, skillfully proved that only a limited number of n flowers constituted the number of n flowers: let an be the number of n petals, that is, an = a1n+a2n+…+. (where 0≤a 1, a2, a3, ..., an≤9), so that 10n- 1≤An≤n×9n, that is, n must satisfy n×9n >；; 10n- 1, that is, (10/9) n
( 10/9)60 = 556.4798…& lt； 600= 10×60，( 10/9)6 1 = 6 18.3 109…>； 6 10= 10×6 1。
For n≥ 6 1, there is (10/9) n >; 10n, therefore, let the inequality (10/9) n.
Spend dozens of girls. Number of flowers per night: 1, 2, 3, 4, 5, 6, 7, 8, 9
Number of chrysanthemums: (20,4,16,37,58,89,145,42)
Number of daffodils: 15337037 1407
Number of peach blossoms:1634,8208,9474
Number of clubs: 54748,92727,93084,0415104 150.
Six, the number of snowflakes: 548834
Number of roses:1741725,4210818,9800817,9926315.
8. Peony numbers: 24678050, 2467805 1, 88593477.
IX. ShandandanNo.: 1465 1 1208, 472335975, 534494836, 9 12985 153.
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Flower number conclusion:
1.n digits (the first digit is not zero). If the sum of the powers of each digit is equal to itself, such a number is called the number of flowers. In order to distinguish it from the following generalized flowers and flowers in a circle, we call such flowers complete flowers.
1, and the maximum number of complete flowers does not exceed 60.
Within 2.60 digits, some complete flowers do not exist. It has been proved that 12, 13, 15, 18, 22, 26, 28, 30 digits do not have complete flowers, and there may be some flowers above 30.
3. The number of complete flowers within 30 has been found, and the maximum number of complete flowers found is 35 digits:12639369517103790328947807201478392.
2. N digits (the first digit can be zero). If the sum of the powers of each digit is equal to itself, such a number is called generalized flower number.
1, and the number of generalized flowers can exceed 60 at most, regardless of the number of digits.
2. The generalized flowers with arbitrary n and n digits do not necessarily exist, but it has been proved that the generalized flowers with 12, 15, 18, 22, 26, 28 and 30 digits do not exist.
3. At present, the number of generalized flowers within 60 has been basically found, and the two largest generalized flowers that have been found are:
56 digits: 0219376224076190839213786089658674401938496187046968.
57 digits: 007425045765382638854534838009896790427088071855172.
Third, the number of flowers in a cyclic circle, we regard the complete number of flowers and the generalized number of flowers as the number of flowers in a cyclic circle with cycle 1 time. Then the general period number is m, which is called the number of flowers in m period. 1 itself is a special number of flowers in 1 cycle. When n is an integer greater than 0:
1. For any number of n digits and powers of n, the number of flowers in a cyclic circle must exist, and at least one circle must exist, for example, n equals 2.
2. For any number of n digits and the power of n, the minimum cycle number (period) (1 is also a special cycle number, except 1) is not necessarily 1 or 2, and it is different for different n, for example, when n = 12.
785119716404 (5 times),
38 12860650 15,
14228 1334933,
35 1 18470 1607,
098840282759,
When n = 18, the minimum cycle is 2. They are:
1878649 19457 18083 1,
375609204308055082,
3. For any number n and the power of n, the maximum period is very small relative to the number n, but it may be tens of millions or even hundreds of millions. The largest circle that has been found exceeds 1 100 million.
4. We also call the number of flowers in the circulating circle the number in the circle or the number on the circle, and the number of flowers in the acyclic circle is also called the number outside the circle. The n power of 1 is also equal to 1. So 1 is the number of flowers of a circle with a cycle number of 1, and it is also the number of circles.
For any N-ary number, N-power, N-ary number can be divided into a group of in-ring numbers and another group of out-of-ring numbers, and all out-of-ring numbers will enter the in-ring numbers after a certain number of N-power operations.