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Thread: Proof Using Modular Arithmetic

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by ilikecandy View Post
    show that if an integer n has at least two distinct odd prime divisors then there exists $\displaystyle k < \varphi(n)$ such that $\displaystyle a^k \equiv 1 \mod n,$ 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.
    $\displaystyle (*)$: suppose $\displaystyle \gcd(u,v)=1, \ r \mid t,$ and $\displaystyle s \mid t.$ if $\displaystyle \ a^r \equiv 1 \mod u$ and $\displaystyle a^s \equiv 1 \mod v,$ then $\displaystyle a^t \equiv 1 \mod uv.$

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

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


    Remark: i suggest you read about the Carmichael $\displaystyle \lambda$ function a little bit to learn about the minimum value of that $\displaystyle k$ in your problem.
    Last edited by NonCommAlg; Oct 19th 2009 at 12:34 AM.
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