Cyclic number

A cyclic number (also: phoenix number ) is a -digit natural number , the product of which when multiplied by a natural number from 1 to contains the same digits as the starting number in the same cyclic order . ${\displaystyle n}$ ${\displaystyle n}$

The cyclic number 142857 multiplied by the numbers 1 to 6

The smallest nontrivial cyclic number in the decimal system is 142857:

${\displaystyle 1\cdot 142.857=142.857}$

${\displaystyle 2\cdot 142.857=285.714}$

${\displaystyle 3\cdot 142.857=428.571}$

${\displaystyle 4\cdot 142.857=571.428}$

${\displaystyle 5\cdot 142.857=714.285}$

${\displaystyle 6\cdot 142.857=857.142}$

${\displaystyle 7\cdot 142.857=999.999}$

Generation

Leonard E. Dickson found that all cyclic numbers are periods of periodic numbers that can be obtained as the reciprocal of certain prime numbers . So is the reciprocal value of 7 is equal to 0.142857142857 ... and contains exactly the first cyclic number as the period: . Such numbers that generate periods of a cyclic number are also called generator numbers: ${\displaystyle {\overline {142857}}}$

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313 ... (episode A001913 in OEIS )

Generator numbers in the decimal system are exactly the prime numbers that meet the following conditions: ${\displaystyle p}$

1. The number base 10 is not a multiple of . ${\displaystyle p}$

2. For natural numbers is not a multiple of . ${\displaystyle 0<n<p-1}$ ${\displaystyle 10^{n}-1}$ ${\displaystyle p}$

3. divides the number , that is, is a multiple of or it applies . ${\displaystyle p}$ ${\displaystyle 10^{p-1}-1}$ ${\displaystyle 10^{p-1}-1}$ ${\displaystyle p}$ ${\displaystyle 10^{p-1}\equiv 1{\pmod {p}}}$

The 486-digit cyclic number that arises at 487 is (so far) the only known number that is itself divisible by its generator number. This means that the period of only has as many digits as that of , just 486 and not the otherwise expected 486 × 487 = 236682. Accordingly, the factor 487 appears in the square when the number is prime factorized with 486 nines or ones ( repunit number ) . ${\displaystyle {\tfrac {1}{487^{2}}}}$ ${\displaystyle {\tfrac {1}{487}}}$

values

Trivial cyclic numbers are all single digit numbers ( ). The first non-trivial cyclic numbers are: ${\displaystyle n=1}$

1. 142857 (6 digits, generated from 1/7)
2. 0588235294117647 (16 digits, generated from 1/17)
3. 052631578947368421 (18 digits, generated from 1/19)
4. 0434782608695652173913 (22 digits, generated from 1/23)
5. 0344827586206896551724137931 (28 digits, generated from 1/29)

properties

• Every non-trivial cyclic number is divisible by 9, e.g. B. 142857/9 = 15873.
• Multiplication by the number of generators gives a sequence of nines, e.g. B. 142857 × 7 = 999999.
• Summing in groups gives a sequence of nines, e.g. B. 142 + 857 = 999 and 14 + 28 + 57 = 99 (Midy's theorem) . The group length must be large enough for this. If the number of digits is not divisible by a number starting with 1, then no more nines sequence is to be expected for a larger number of groups.
• The proportion of the generator numbers in the set of all prime numbers is the Artin constant C = 0.3739558136192… (sequence A005596 in OEIS ). This is linked to the Primzeta function via the Lucas numbers and can be determined.

Other number bases

Cyclic numbers can be formed in almost all number systems as long as their number base is not a square number; in the quaternary system (base 4 = 2²) or in the hexadecimal system (base 16 = 4²) there are therefore no cyclic numbers.

Example: Cyclic number in the binary system

1. 0001011101 × 0001 = 0001011101
2. 0001011101 × 0010 = 0010111010
3. 0001011101 × 0011 = 0100010111
4. 0001011101 × 0100 = 0101110100
5. 0001011101 × 0101 = 0111010001
6. 0001011101 × 0110 = 1000101110
7. 0001011101 × 0111 = 1010001011
8. 0001011101 × 1000 = 1011101000
9. 0001011101 × 1001 = 1101000101
10. 0001011101 × 1010 = 1110100010
11. 0001011101 × 1011 = 1111111111

In many number bases you can represent cyclic numbers according to the formula (with the number base and the divisor ), provided and ( ) are relatively prime and the number of modules ( modulo ) is not , or is greater . Nice cyclic numbers contain each digit only once. Here is a recent table: ${\displaystyle z}$ ${\displaystyle z={\frac {B^{q-1}-1}{q}}}$ ${\displaystyle B}$ ${\displaystyle q}$ ${\displaystyle B}$ ${\displaystyle q}$ ${\displaystyle 0<q<B}$ ${\displaystyle B}$ ${\displaystyle q}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 3}$

1. ${\displaystyle B=5}$, : 13 ${\displaystyle q=3}$
2. ${\displaystyle B=7}$, : 1254 ${\displaystyle q=5}$
3. ${\displaystyle B=8}$, : 1463 ${\displaystyle q=5}$
4. ${\displaystyle B=10}$, : 142857 ${\displaystyle q=7}$
5. ${\displaystyle B=12}$, : 186A35 ${\displaystyle q=7}$
6. ${\displaystyle B=13}$, : 12495BA837 ${\displaystyle q=11}$
7. ${\displaystyle B=20}$, : 13ABF5HCIG984E27 ${\displaystyle q=17}$
8. ${\displaystyle B=22}$, : 13A95H826KIBCG4DJF ${\displaystyle q=19}$
9. ${\displaystyle B=31}$, : 248H36CPK9J7ETSQMDROI5ALBNG ${\displaystyle q=29}$
10. ${\displaystyle B=32}$, : 139TPC4D7N5GHKUSM26JRIO8QFEB ${\displaystyle q=29}$
11. ${\displaystyle B=34}$, : 139TKSHILVRE8QAWUO4D5GFC26JP7N ${\displaystyle q=31}$
12. ${\displaystyle B=39}$, : 1248GXSHZWQDRFVO9IbaYUM5AL36CPBN7ETK ${\displaystyle q=37}$
13. ${\displaystyle B=46}$, : 139SeThdQYAW4CcNORbKEigaH5G26JBZDfX7MLI8PV ${\displaystyle q=43}$
14. ${\displaystyle B=50}$, : 139Sa8PQTdI4CcEiY26J7MH139Sa8PQTdI4CcEiY26J7MH ${\displaystyle q=47}$
15. ${\displaystyle B=56}$, : 139STWgEiL7MAVd5FlUZphHrnasqkRQNDfBYmXjOGoe8PK4Cc26J ${\displaystyle q=53}$
16. ${\displaystyle B=63}$, : 1248GX36COna9IbBMjRtmY5AKfJdFUzywskTxuocDQriPpeHZ7ESvqgLhNlW ${\displaystyle q=61}$

The letters A, B, C, ... are used for the numerical values ​​10, 11, 12, ... and a, b, c, ... for the numerical values ​​36, 37, 38, ...

literature

• Manfred Scholtyssek: Hexeneinmaleins , 3rd edition 1984, children's book publisher Berlin (GDR)
• Leonard E. Dickson : History of the Theory of Numbers . Washington 1932 (3 vol.)

Web links

• Eric W. Weisstein : Cyclic Number . In: MathWorld (English).

Individual evidence

1. ↑ Endre Hódi (Ed.): Mathematisches Mosaik , Urania, Leipzig 1977
2. ↑ Manfred Scholtyssek: Hexeneinmaleins , 3rd edition 1984, children's book publisher Berlin (GDR)
3. ↑ Eric W. Weisstein : Full Reptend Prime . In: MathWorld (English).
4. ↑ Factorizations of 11… 11 (Repunit). ( Memento from November 12, 2013 in the Internet Archive )
5. ↑ Eric W. Weisstein : Midy's Theorem . In: MathWorld (English).