High composite number

The title of this article is ambiguous. To use for numbers that have at most 2, 3, 5, 7 as prime factors, see 7-even number .

A highly composite number ( Engl. Highly composite number , in short: HCN ) is a positive integer , the more divider has smaller than any positive integer. Due to their maximum divisibility, such numbers are a kind of counterpart to the prime numbers . The Indian mathematician Srinivasa Ramanujan was one of the first to investigate these numbers and their properties in more detail and published an extensive article on them in 1915.

The first twenty highly compounded numbers

Running index ${\displaystyle k}$	Follow in OEIS	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16	17th	18th	19th	20th
${\displaystyle k}$-th composite number	A002182	1	2	4th	6th	12	24	36	48	60	120	180	240	360	720	840	1260	1680	2520	5040	7560
Number of divisors	A002183	1	2	3	4th	6th	8th	9	10	12	16	18th	20th	24	30th	32	36	40	48	60	64

properties

construction

Two necessary properties of highly composed numbers result from the number function . As the fundamental theorem of arithmetic says, every positive natural number is structured as follows: ${\displaystyle n}$

${\displaystyle n=p_{1}^{c_{1}}\cdot p_{2}^{c_{2}}\cdot \ldots \cdot p_{k}^{c_{k}}}$    and     ${\displaystyle p_{1}<p_{2}<\dotsb <p_{k},}$

where are the prime numbers. The exponents are natural numbers other than zero . For the empty product results . The definition of the number function then provides the number of divisors for natural numbers : ${\displaystyle p_{i}}$ ${\displaystyle c_{i}}$ ${\displaystyle k=0}$ ${\displaystyle n=1}$ ${\displaystyle d(n)}$

${\displaystyle d(n)=(c_{1}+1)\cdot (c_{2}+1)\cdot \ldots \cdot (c_{k}+1)}$.

For highly composite numbers it follows from this formula:

• The prime numbers are exactly the first prime numbers, because every omitted prime number would make it possible to construct a smaller one with the same number of divisors. ${\displaystyle k}$ ${\displaystyle p_{i}}$ ${\displaystyle k}$ ${\displaystyle n}$
• The sequence of the exponents is descending, it applies . Otherwise it would be possible by interchanging exponents to construct a smaller one with the same number of divisors. ${\displaystyle c_{1}\geq c_{2}\geq \dotsb \geq c_{k}}$ ${\displaystyle n}$

These two properties are necessary, but not sufficient. So, excluding and , must be the last exponent . ${\displaystyle n=4}$ ${\displaystyle n=36}$ ${\displaystyle 1}$

Example:

${\displaystyle 720=2^{4}\cdot 3^{2}\cdot 5^{1}}$has dividers. That’s more factors than any smaller number. So is a highly composite number. ${\displaystyle (4+1)\cdot (2+1)\cdot (1+1)=30}$ ${\displaystyle 720}$

Applications

The property of having as many dividers as possible offers practical advantages and is therefore often sought deliberately. The angular degree system of 360 ° is based on a highly composite number. The hours of 24, minutes and seconds of 60 units as well as the old coin system of Charlemagne with the relation one pound of silver equals 240 pfennigs or denarii are to be mentioned here. In Prussia from 1821 to 1873 a thaler was equal to 360 pfennigs.

If you develop a scale or circle division on the basis of a highly complex number, this scale can be evenly divided in a particularly large number of different ways.

Ramanujan and highly compound numbers

The Indian Srinivasa Ramanujan (1887–1920) was one of the first mathematicians to deal extensively with highly composed numbers. In doing so, he found the rule of non-increasing exponents mentioned above. The rule can be used to construct highly composite numbers. Ramanujan himself made a list of over a hundred of the first highly composed numbers. But he overlooked a single one, namely the number 293,318,625,600. Today you can find online lists with over a hundred thousand numbers in this sequence .

literature

• S. Ramanujan: Highly composite numbers . In: Proc. London. Math. Soc . tape 14 , 1915, p. 347-409 ( ramanujan.sirinudi.org [PDF] ( Review. In: Zentralblatt Math)).
• József Sándor, Dragoslav S. Mitrinović, Borislav Crstici: Handbook of Number Theory I . Springer-Verlag, Dordrecht NL 2006, ISBN 978-1-4020-4215-7 , p. 45-46 .
• Paul Erdős : On Highly Composite Numbers (PDF) In: Journal of the London Mathematical Society , 1944.
• Paul Erdős, L. Alaolglu: On Highly Composite and Similar Numbers (PDF) In: Transaction of the Americal Mathematical Society , Vol. 56, No 3, November 1944, pp. 448–469.
• Srinivasa Ramanujan , Jean-Louis Nicolas, Guy Robin: Highly Composite Numbers (PDF) In: The Ramanujan Journal , I, 1997, pp. 119–153.

Web links

• Eric W. Weisstein : Highly Composite Number . In: MathWorld (English).
• Achim Flammenkamp: Further information , Bielefeld University.
• Special highly compound numbers
• Associated site of Michigan State University

Individual evidence

1. ↑ “ They are as unlike a prime as a number can be. ”- Hardy , after Robert Kanigal: The Man Who Knew Infinity: A Life of the Genius Ramanujan . Scribner, New York 1991, p. 232.
2. ↑ Eric W. Weisstein : Highly Composite Number . In: MathWorld (English).