Cross product
The cross product of a natural number is - similar to the cross sum - the product of its numerical values . The decimal cross product of 5496 is, for example, 5 * 4 * 9 * 6 = 1080. Just like the cross sum, the cross product also depends on the number system used . In the respective number system, single-digit numbers correspond to their own cross product.
Graph course
The graph of the cross product function, which assigns its cross product q (n) to every natural number n, has a characteristic curve. It consists of successive spikes that reach ever higher peaks. Between these spikes q (n) always falls to 0; namely whenever there is at least one digit 0 in n.
This behavior occurs in every power of ten - the range 0 ≤ n ≤ 10 forms a point just like 0 ≤ n ≤ 10,000. In this way, self-similarity appears in the graph of q (n) . When considering a power of ten, the first two points are always the same size, the following eight represent two, three, four times, etc. of the first points.
The smallest function value q (n) is 0, there is no upper limit.
Iterated cross product
If you generate a number sequence in which each number is the cross product of its predecessor, the sequence ends for each multi-digit starting number after a finite number of steps with a single-digit number. This is due to the fact that the cross product of a multi-digit number is always smaller than the number itself.
3784 → 3 7 8 4 = 672 → 6 7 2 = 84 → 8 4 = 32 → 3 2 = 6
75664 → 7 5 6 6 4 = 5040 → 5 0 4 0 = 0
The number of necessary steps is referred to as persistence (engl. Multiplicative persistence ) a number. Thus 3784 has the persistence 4 and 75664 the persistence 2. The one-digit number that is obtained at the end of the chain is called the multiplicative digital root (German: "multiplicative digit root ").
For the following persistence, the smallest starting numbers are known in the decimal system (sequence A003001 in OEIS ). A number with a persistence of 12 is not yet known.
| Persistence of n | Smallest number n |
|---|---|
| 1 | 10 |
| 2 | 25th |
| 3 | 39 |
| 4th | 77 |
| 5 | 679 |
| 6th | 6 788 |
| 7th | 68 889 |
| 8th | 2,677,889 |
| 9 | 26 888 999 |
| 10 | 3 778 888 999 |
| 11 | 277 777 788 888 899 |
literature
- Eric Milou, Jav L. Schiffman: The Spirit of Discovery: The Digital Roots of Integers . In: Mathematics Teacher. Volume 101 No. 5, December 2007, pp. 379-383.
- Richard K. Guy: Unsolved Problems in Number Theory . 3. Edition. Springer-Verlag, 2004, ISBN 0-387-20860-7 , p. 399 (problem F25).
- NJA Sloane: The Persistence of a Number . In: Journal of Recreational Mathematics , Volume 6, No. 2, 1973, pp. 97-98.
- Clifford A. Pickover : Dr. Googol's wondrous world of numbers . Heinrich Hugendubel Verlag, 2002, ISBN 3-423-34177-7 , Chapter 9: Stubborn numbers .
Web links
- Follow A007954 in OEIS (cross products)
- Sequence A003001 in OEIS (smallest starting numbers for persistence)
- Eric W. Weisstein : Multiplicative Digital Root . In: MathWorld (English).
- Sloane's conjecture on multiplicative digital root . In: PlanetMath .
Individual evidence
- ^ Jens Fleckenstein, Walter Fricke, Boris Georgi: Excel - the riddle book . Pearson Education 2007 ISBN 3-8272-4244-4 , ( online limited version (Google Books) )
- ↑ Multiplicative digital root on PlanetMath
- ↑ Pickover (see literature )