# Thread: Difficult Congruence related questions

1. ## 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?

2. Originally Posted by jessicafreeman
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 $\displaystyle (6k+1)(6m+1)$ is again of the form $\displaystyle 6t+1$ , 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 $\displaystyle p_1,\ldots,p_r$ are all the primes of that form, consider $\displaystyle M:=(p_1\cdot\ldots\cdot p_r)^2+5$

(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