Results 1 to 2 of 2

Math Help - Proof Using Modular Arithmetic

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    14

    Proof Using Modular Arithmetic

    show that if an integer n has at least two distinct odd prime divisors then there exists k<φ(n) such that
    a^k ≡ 1(mod n)
    for every a relatively prime to n.


    What I have so far: The expression a^k ≡ 1(mod n) looks similar to Euler's Generalization of Fermat's Little Theorem. Also, n can be expressed as xp1p2, where x is a positive integer, and p1 and p2 are odd distinct prime numbers.

    I know this isn't much, but I don't to where to go afterwards to complete the proof.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by ilikecandy View Post
    show that if an integer n has at least two distinct odd prime divisors then there exists k < \varphi(n) such that a^k \equiv 1 \mod n, for every a relatively prime to n.

    What I have so far: The expression a^k ≡ 1(mod n) looks similar to Euler's Generalization of Fermat's Little Theorem. Also, n can be expressed as xp1p2, where x is a positive integer, and p1 and p2 are odd distinct prime numbers.

    I know this isn't much, but I don't to where to go afterwards to complete the proof.
    (*): suppose \gcd(u,v)=1, \ r \mid t, and s \mid t. if \ a^r \equiv 1 \mod u and a^s \equiv 1 \mod v, then a^t \equiv 1 \mod uv.

    now let n=p^{\alpha} q^{\beta}m, where p \neq q are some odd primes and \gcd(m,p)=\gcd(m,q)=1. let t=\text{lcm}(\varphi(p^{\alpha}), \varphi(q^{\beta})) < \varphi(p^{\alpha})\varphi(q^{\beta}), where \varphi is the Euler's totient function.

    let k=t \varphi(m) < \varphi(n) and suppose \gcd(a,n)=1. by (*) we have a^t \equiv 1 \mod p^{\alpha}q^{\beta}. we also have a^{\varphi(m)} \equiv 1 \mod m. so, again by (*), we get: a^k \equiv 1 \mod n. \ \Box


    Remark: i suggest you read about the Carmichael \lambda function a little bit to learn about the minimum value of that k in your problem.
    Last edited by NonCommAlg; October 19th 2009 at 01:34 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. modular arithmetic proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 8th 2010, 06:03 AM
  2. Modular arithmetic HELP!!
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: May 16th 2009, 01:02 PM
  3. Modular Arithmetic
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 17th 2009, 04:57 PM
  4. Modular arithmetic help
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: December 4th 2008, 09:47 AM
  5. Modular Arithmetic
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: October 15th 2006, 08:07 PM

Search Tags


/mathhelpforum @mathhelpforum