Hi!
My name is Jesule and I am a student. I am stumped, and I do not Know how to solve the following problem:
Given that p, p+10 and p+14 are prime numbers, find p.
Please help. I have no idea how to start.
Hi Jesule,
Hi have two responses to your question. One from Claude.
Similar question: Given that p, p + 2, p + 6, p + 8 and p + 14 are all prime, find p.
Step 1: Hmmmm, 5 fits the bill, since 5, 7, 11, 13 and 19 are all prime.
Step 2: Are there other possibilities? Well, if p leaves a remainder of x when divided by 5, then the remainders of p + 2, p + 6, p + 8 and p + 14 when divided by 5 will be x+2, x+1, x+3 and x+4 modulo 5. So, one of p, p + 2, p + 6, p + 8 and p + 14 has to be a multiple of 5, and since they are all prime, one of them has to be 5. We have seen that setting p = 5 works; setting p + 2 = 5 implies that p + 6 = 9, which is not prime. And of course, p + 6, p + 8 and p + 14 cannot be 5, so p = 5 is the only solution.
Claude
and a second from Chris
Try working modulo 3. The middle prime has to be either 1 or 2 mod 3. Add 4 to that (mod 3) and subtract 10 mod 3. You'll find if the middle guy is prime then at most one of the others could be.
Chris