In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, an ''n''-Knödel number for a given
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
''n'' is a
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
''m'' with the property that each ''i'' < ''m''
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to ''m'' satisfies
.
The concept is named after
Walter Knödel
Walter Knödel (May 20, 1926 – October 19, 2018) was an Austrian mathematician and computer scientist. He was a computer science professor at the University of Stuttgart.
Born in Vienna, Walter Knödel studied mathematics and physics at the Un ...
.
The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all ''n''-Knödel numbers is denoted ''K''
''n''.
The
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of .Brown, James Robert.� ...
''K''
1 is the
Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
s.
There are infinitely many ''n''-Knödel numbers for a given ''n''.
Due to
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, then a^ is congruent to 1 modulo , where \varphi denotes Euler's totient function; that ...
every composite number ''m'' is an ''n''-Knödel number for
where
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
.
Examples
References
Literature
*
*
Eponymous numbers in mathematics
Number theory
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