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Math Help - Difficult Congruence related questions

  1. #1
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    Difficult Congruence related questions

    hi there.

    could anyone give me a hand with these ;3. xoxox

    (1) Show that if you multiply 2 integers of the form 6j + 1 together, you get another integer of that form.

    (2) Show that there are infinitely many primes of the form 6j + 5.
    This is the hint: Try modifying the proof that there are infinitely many primes. Instead of considering p1 . . . pn + 1, consider some other integer constructed from p1 . . . pn that will certainly be congruent to 5(mod 6).

    (3) Show that if p and p + 2 are both prime and p >= 5, then 6 must divide p + 1.
    This is the hint: What congruence classes modulo 6 can p and p + 2 belong to?
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  2. #2
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    Quote Originally Posted by jessicafreeman View Post
    hi there.

    could anyone give me a hand with these ;3. xoxox

    (1) Show that if you multiply 2 integers of the form 6j + 1 together, you get another integer of that form.


    Do it!!! Prove that  (6k+1)(6m+1) is again of the form 6t+1 , for some integer t...!

    (2) Show that there are infinitely many primes of the form 6j + 5.
    This is the hint: Try modifying the proof that there are infinitely many primes. Instead of considering p1 . . . pn + 1, consider some other integer constructed from p1 . . . pn that will certainly be congruent to 5(mod 6).


    If p_1,\ldots,p_r are all the primes of that form, consider M:=(p_1\cdot\ldots\cdot p_r)^2+5



    (3) Show that if p and p + 2 are both prime and p >= 5, then 6 must divide p + 1.
    This is the hint: What congruence classes modulo 6 can p and p + 2 belong to?

    Answer the hint and then think

    Tonio
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