Expressions of the Smarandache Coprime Function

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:

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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.

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