Liquor number

A schnapps number ( listen ? / I ) is a multi-digit natural number that is only represented by identical digits . In mathematics , these numbers are also known as Repdigit , English for repeated digits , German repeated digits .

Examples and definition

Examples of liquor numbers are:

• 11
• 666
• 3333

All liquor numbers are of shape

${\ displaystyle d \ sum _ {k = 0} ^ {n-1} b ^ {k} = d {\ frac {b ^ {n} -1} {b-1}}}$,

where is the digit used, the number of digits and the base used. ${\ displaystyle d}$${\ displaystyle n \ geq 2}$${\ displaystyle b}$

Origin of the name

The name is derived from games with several participants, in which the course manifests itself as the result of an addition that is logged . If the total score of one of the players reaches a schnapps number, free drinks - for example a schnapps - may be due for the other players depending on the existing rules of the game or verbal agreements .

Another interpretation relates to the fact that double vision can occur after excessive alcohol consumption , which can turn a 2 into a 22 or a 33 into a 333 or a 3333.

Use of liquor numbers

In addition to the above-mentioned use in drinking games, there are other areas in which liquor numbers play a special role:

Since the equality of the digits depends arbitrarily on the selected number base (here in the example decimal ), it is a mild form of number magic . The schnapps number 666, which is referred to as the number of the beast in the Revelation of John , is of particular importance . It is used particularly in the heavy metal environment , for example in the song The Number of the Beast by Iron Maiden .

In mathematics, the repdigits in the dual system play an important role (see also Mersenne prime ). In this place value system , repdigits can only consist of the number 1 and are therefore called repunit (repeated unit). Regardless of the place value system used, among the repdigits only the repunits with a prime number of digits can be prime numbers , all other repdigit numbers are compound.

Different meaning

Notwithstanding the above definition, axially symmetrical sequences of digits are also occasionally referred to as schnapps numbers. They are then number palindromes , for example:

• 121
• 9889
• 10001

Numbers that have the same value when upside down (by rotating in the plane of the drawing around the center point) are occasionally referred to as schnapps numbers, for example:

• 69
• 609
• 9886