An Extension of the Euler Phi-function to Sets of Integers Relatively Prime to 30


  •  Mbakiso Mothebe    
  •  Ben T. Modise    

Abstract

Let $n \geq 1$ be an integer and let $S= \{1,7,11,13,17,19,23,29\},$ the set of integers which are both less than and relatively prime to $30.$ Define $\phi_3(n)$ to be the number of integers $x, \; 0 \leq x \leq n-1,$ for which $\gcd(30n, 30x+i) = 1$ for all $i \in S.$ In this note we show that  $\phi_3$ is multiplicative, that is, if $\gcd(m, n)=1,$ then $\phi_3(mn)=\phi_3(m)\phi_3(n).$ We make a conjecture about primes generated by S.


This work is licensed under a Creative Commons Attribution 4.0 License.
  • ISSN(Print): 1916-9795
  • ISSN(Online): 1916-9809
  • Started: 2009
  • Frequency: bimonthly

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