Expressions of the Smarandache Coprime Function Sebastian Martin Ruiz, Avda de RegIa, 43 Chipiona, 11550 Cadiz, Spain
Smarandache Coprime Function is defined this way:
_{ 0 if C k (nl,n2,···,nk ) 1
nl,n2,···,nkarecoprimenumbers otherwise
\Ye see two expressions of the Smarandache Coprime Function for k=2. EXPRESSION 1:
C( 2 nl,
n2 )
=-
E[
nln2-tcm(nl,n2)] nln2
E( x) = the biggest integer number smaller or equal than x. If nl, n2 are coprime numbers:
If nl, n2 aren't coprime numbers:
62
EXPRESSION 2:
II
II
d I nl
d' I n2 d' > 1
d> 1
If nl, n2 are coprime numbers then d
II
II
Id-d'i
=I d' V d, d' =I 1
Id-d'i
d I nl d' I n2 d> 1 d' > 1 :::} 0 < --=-=:::------ < 1 :::} C 2 ( n 1, n2) = 0 II(d+d') din, din,
II
Ifnl,n2 aren't coprime numbers 3d= d' d> I,d' Smarandache coprime function for k 2: 2.
> I:::} C 2(nl,n2) = 1
If nl , n2, ... ,nl; are coprime numbers:
If nl, n2,···, nl; aren't coprime numbers: GCD(nl, n2,···, nl;) > 1
References: 1. E. Burton, "Smarandache Prime and Coprime Functions"
2. F. Smarandache, "Collected Papers", Vol. II, 200 p.,p. 137, Kishinev University Press.
63