Factorization of integers

  1. Why is factorization of integers important on a first number theory course? Where is factorization used in real life? Are there examples which have a real impact? I am looking for examples which will motivate students.
  2. jcsd
  3. SteamKing

    SteamKing 10,972
    Staff Emeritus
    Science Advisor
    Homework Helper

    I'll take a stab.
    Factorization helps determine if a given integer is prime, and one use for prime integers is in devising cryptography keys, which are used quite a bit for, among other things, encrypting sensitive data which might be swapped around on the internet. (NSA, how'm I doin' so far?)

    If you have an arbitrary integer of n-digits, how long does it take to determine the factors (if any) of this integer? That's a pretty basic question for number theory to answer. Is it a couple of hours, a couple of days, a couple of years, a couple of centuries, or what? Can a better (= quicker) algorithm be devised?

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  4. Stephen Tashi

    Stephen Tashi 4,470
    Science Advisor
    2014 Award

    Suppose we can motivate an interest in Diophantine equations. Their solution entails finding greatest common divisors. Would that also lead in a natural way to focusing on prime numbers?
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  5. PeroK

    PeroK 1,693
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    Gold Member

    Much of Internet security uses Public Key Cryptography, which depends on large integer factorisation. See, for example:

  6. If you want to solve a quadratic equation by factorisation the you need to be able to factorises integers.
    That is to solve

    ax2 + bx + c = 0

    you need to factorises a and c.
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