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Math Help - Proofs a bout primes

  1. #1
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    Unhappy Proofs a bout primes

    Here are some proofs re primes I need help with

    1. Prove that any prime of the form 3n+1 is also of the form 6m+1

    2. Show that any integer of the form 3n+2 must have a prime factor of the same form.

    3. Show that 7 is the only prime of the form n^3-1

    4. Show that p=5 is the only prime for which 3p+1 is a perfect square.

    5. If p is prime and p divides a^n, show that p^n divides a^n

    6. Show that every integer of the form n^4+4 is composite

    Any help gratefully accepted.
    Cheers
    Cabouli
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  2. #2
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    Quote Originally Posted by Cabouli View Post
    1. Prove that any prime of the form 3n+1 is also of the form 6m+1
    A prime number (excluding 2 and 3) must have form 6k+1 or 6k+5. Let p have form 3k+1. If p had form 6k+5 then 6k+5=3(2k+1)+2 so p would have form 3k+2, a contradiction. Thus, p must have form 3k+1.

    2. Show that any integer of the form 3n+2 must have a prime factor of the same form.
    Let x be a positive integer of the form 3k+2. Since it is odd we know x is a product of odd prime numbers. Thus, each prime factor has form 3k+1 or 3k+2. If each prime factor had form 3k+1 then the product of all of them still will have form 3k+1, which is a contradiction.
    3. Show that 7 is the only prime of the form n^3-1
    Hint: n^3 - 1 = (n-1)(n^2+n+1)
    4. Show that p=5 is the only prime for which 3p+1 is a perfect square.
    We will asume that p\geq 3. If 3p+1=n^2 \implies 3p = (n-1)(n+1) by unique factorization it forces n-1 =3\text{ and }n+1=p. Thus, p=5 is the solution.
    5. If p is prime and p divides a^n, show that p^n divides a^n
    If p|a^n then p|a then p^n|a^n.
    6. Show that every integer of the form n^4+4 is composite
    See this.
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  3. #3
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    Quote Originally Posted by Cabouli View Post
    1. Prove that any prime of the form 3n+1 is also of the form 6m+1
    2 is the only even prime and its not of the form 3n+1

    hence all primes of the form 3n+1 must be odd

    this means that n must be even so n = 2m
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