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Larson 4.1.6

  1. Jan 7, 2008 #1
    [SOLVED] Larson 4.1.6

    1. The problem statement, all variables and given/known data
    Prove that there are infinitely many natural numbers a with the following property: The number n^4+a is not prime for any number n.

    2. Relevant equations

    3. The attempt at a solution
    I cannot even think of one such natural number a. :(
    I need to find some way to factor this after we put some restrictions on a. That is we need to express a in a special form that makes this factorable. If a is equal to b^4, it is not necessarily factorable. In fact, I don't know of any power of b that will make it factorable. a cannot be a function of n. I really don't know what to do.
  2. jcsd
  3. Jan 7, 2008 #2
    a=(multiple of 5)-1
    hence infinite
    i guess
    correct me if iam wrong
    have you any idea of fermat theorm
    n^5-n is divisible by 5
    can be prooved ,it is a simpler form of fermat theorm
    n(n^4-1) certainly n^4 -1 is divisible by 5
    add any multiple of 5 to it
    you get
  4. Jan 7, 2008 #3
    Fermat's Little Theorem says that if p is a prime number that does not divide an integer n, then [itex] n^{p-1} \equiv 1 \mod p [/itex].

    Therefore that will only apply when n is not divisible by 5. We need a proof for all n in N.
  5. Jan 7, 2008 #4
    Maybe you can use Sophie Germain's identity:
    [tex]a^4 + 4 b^4 = (a^2 + 2 b^2 + 2 a b) (a^2 + 2 b^2 - 2 a b)[/tex].
  6. Jan 7, 2008 #5
    Wow. Thanks. I'm glad I posted this question because I never would have thought of that.
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