1. The problem statement, all variables and given/known data Prove that 6k+5 produces an infinite amount of primes. k is an integer 3. The attempt at a solution We first observe that 6k+1,6k+3,6k+5 produce all the odd integers. Next we see that (6k+1)(6k'+1)=6(6kk'+k+k')+1 so the product of integers of the form (6k+1)(6k'+1)=6k''+1. And we see that 6k+3 is divisible by 3 so it never produces primes. when k=1 we see that 6+5=11 and when k=2 we get 12+5=17. so we know that 6k+5 does produce primes. Now let [itex]X= p_1,p_2,.......p_n [/itex] be the set of all primes of the form 6k+5. Now we construct [itex]Y= 6(p_1*p_2*.....*p_n)+5 [/itex] Now this new number is bigger than any [itex] p_n [/itex] And lets assume that X is finite. And lets assume 5 is not in X. But no [itex] p_n [/itex] divides Y so either Y itself is prime or there is a larger prime of the form 6k+5 that divides it. there are an infinite amount of primes of the form 6k+5.