# Is is true that $p_{n+1}p_n \mod p_{n+1}+p_n$ is prime?

I noticed a pattern when messing around with prime numbers, and I conjectured that, if $p_n$ is the nth prime number, then $$(p_{n+1}p_n)\mod(p_{n+1}+p_n)$$ is also prime. I have very little experience in dealing with prime numbers, so I'm not even sure how to start this proof. Can someone please either show me how to start this or disprove it? It would also be helpful if you know of any online resources that have any kind of practice proofs regarding prime numbers.

Thanks!

$$17 \times 19 = 323 \equiv 35 \pmod { 36}$$

• Ah, I see. Had I gone out one case farther, I would have caught that one. Can you recommend any good places where I can find practice proofs about primes? – Franklin Pezzuti Dyer May 12 '17 at 19:54
• @Frpzzd it's hard to recommend it without knowing your background. A traditional source to follow (assuming you are interested in a more analytic approach) is Apostol's number theory book. It covers the basics and it has exercises. You can probably find it online. – Alex Macedo May 12 '17 at 20:03
• Okay, thanks! I'll take a look at it. – Franklin Pezzuti Dyer May 12 '17 at 20:08

$$47\times 53 = 2491 =_{100} 91$$

Edit: oh well Alex beat me to it

This fails immediately:

$$2\cdot 3 = 6\equiv 1\mod 5.$$

• Well, I chose to exclude that case because the primality of $1$ is disputed. – Franklin Pezzuti Dyer May 12 '17 at 20:37
• @Frpzzd The primality of 1 is not disputed, but if you only want to include non-units that's also a legitimate course of action, but you should say so. – Adam Hughes May 12 '17 at 20:38
• That's not what I was thinking, but that's a valid reason as well. It isn't of much importance anyways, since there are other counterexamples. – Franklin Pezzuti Dyer May 12 '17 at 20:42
• @Frpzzd yes certainly, I just took the occam's razor approach since it's a nice, easily hand-verified counterexample. :) – Adam Hughes May 12 '17 at 20:43