Euler totient function φ(n)

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:arithmetic_function
gptkbp:alsoKnownAs	gptkb:Euler's_totient_function phi function
gptkbp:application	gptkb:RSA_cryptosystem gptkb:Euler's_theorem number theory
gptkbp:codomain	non-negative integers
gptkbp:defines	number of positive integers less than or equal to n that are coprime to n
gptkbp:designer	no simple closed form
gptkbp:domain	positive integers
gptkbp:form	φ(n) = n × Π(1 - 1/p) over all distinct prime divisors p of n
gptkbp:hasSpecialCase	φ(1) = 1 φ(p) = p-1 for prime p
gptkbp:introduced	gptkb:Leonhard_Euler
gptkbp:introducedIn	1763
gptkbp:namedAfter	gptkb:Leonhard_Euler
gptkbp:notation	φ(n) phi(n)
gptkbp:property	sum of φ(d) over all divisors d of n equals n φ(mn) = φ(m)φ(n) if gcd(m, n) = 1 multiplicative function average order is 6n/π²
gptkbp:relatedTo	gptkb:Carmichael_function gptkb:Möbius_function prime numbers
gptkbp:sequence	gptkb:A000010_(OEIS)
gptkbp:usedIn	cryptography group theory modular arithmetic
gptkbp:bfsParent	gptkb:A002322
gptkbp:bfsLayer	8
https://www.w3.org/2000/01/rdf-schema#label	Euler totient function φ(n)