One thousand and one

This article describes the natural number. For the card game, see Thousand and One , for the year 1001 .

One thousand and one
1001
presentation
Roman	M I
dual	11 1110 1001
Octal	1751
Duodecimal	6B5
Hexadecimal	3E9
Morse code	- - - - - - - - - - - - - - - - - - -
Mathematical properties
sign	positive
parity	odd
Factorization	${\displaystyle 7\cdot 11\cdot 13}$
Divider	1, 7, 11, 13, 77, 91, 143, 1001

The number one thousand one (1001), sometimes thousand and one , or shortly thousand one written, which is natural number between 1000 and 1002. It is odd . 1001 is not a prime number , but the product of three prime numbers:

${\displaystyle 1001=7\cdot 11\cdot 13}$

Divisibility rule

A natural number is divisible by 7 (or 11 or 13) if the alternating sum of the groups of three formed from the right is divisible by 7 (or 11 or 13).

As an example, consider 47.405.100.203.526. The alternating sum of the groups of three is 526−203 + 100−405 + 47 = 65. Since 65 is divisible by 13, the starting number must also be divisible by 13. Since 65 is neither divisible by 7 nor by 11, the starting number cannot be divisible by 7 or 11 either.

The reason for this rule is that

${\displaystyle 1001=7\cdot 11\cdot 13}$   and   ${\displaystyle 1000\equiv -1\mod 1001}$

where mod stands for modulo . In the example above, modulo 1001 applies

${\displaystyle 47.405.100.203.526=526\cdot 1000^{0}+203\cdot 1000^{1}+100\cdot 1000^{2}+405\cdot 1000^{3}+47\cdot 1000^{4}}$

${\displaystyle \equiv 526\cdot (-1)^{0}+203\cdot (-1)^{1}+100\cdot (-1)^{2}+405\cdot (-1)^{3}+47\cdot (-1)^{4}=526-203+100-405+47=65}$

and this results in the divisibility claim.

In particular, every six-digit number with the sequence ABCABC, where A or B can also be a zero and the number can therefore be four or five-digit, is divisible by 1001 and thus by 7, 11 and 13, because in this case the alternating sum of the groups of three is equals ABC − ABC = 0.

Other mathematical properties

The number 1001 is

• a pentagonal number (pentagonal number),
• a pentatope number (hypertenrahedral number),
• a sphenic number ,
• a palindrome of numbers .

Non-mathematical meanings

The number 1001 is often used to make a statement in the sense of "a lot and a little more" or "more than you think". This is especially true in advertising. There are game collections with 1001 games - here mostly many dice games and their variations are counted - collections with 1001 font files (fonts), offers with 1001 books, songs, etc.

The title of the oriental collection of stories, Arabian Nights , also means “very long”, and correspondingly a thousand and one days .

In the RAL color system , 1001 corresponds to beige .

Individual evidence

1. ↑ Guido Walz (Ed.): Lexicon of Mathematics. Volume 5, page 191: Divisibility criteria.