# Can 4 distinct prime numbers be related in such a way?

1. Jul 12, 2013

### JFGariepy

Hi everyone,

I've been bumping on this problem for a while and wondered if any of you had any clue on how to approach it. My question is whether the following equality is possible for 4 distinct prime numbers :

PxPy + Pw = PwPz + Px

where Px, Py, Pw, Pz are odd prime numbers, and each of them is unique, thus not equal to the others.

If impossible, how would you prove that ?

Thanks!

JF

2. Jul 12, 2013

### economicsnerd

An equivalent way of phrasing the question:

Do there exist k,p,q with p,q distinct primes such that kp+1, kq+1 are also primes?

[My p,q correspond to your Px,Pw.]

3. Jul 12, 2013

### economicsnerd

Oh, it's not exactly equivalent. If p<q, then I need to further require that kp+1 is distinct from q.

4. Jul 12, 2013

### Staff: Mentor

How is that equivalent?
With k=2, there are many such primes p,q. Examples: p=3, q=5.
I don't see how we can construct the equality from the first post based on them.

This is possible with other even k, too. Your k cannot be odd, otherwise kp+1 and kq+1 would be even and not prime.

PxPy + Pw = PwPz + Px can be written as Px(Py-1) = Pw(Pz-1)
As Px and Pw are distinct primes, (Pz-1) has to be divisible by Px and (Py-1) has to be divisible by Pw. As both (Pz-1) and (Py-1) are even, Pz>2Px and Py>2Pw.

5. Jul 12, 2013

### JFGariepy

Interesting.

I also would add the following conditions:

Since the equality Px(Py-1) = Pw(Pz-1) needs to reflect 1 integer that has a unique prime factorization, (Py-1) is an even number with prime factor Pw and (Pz-1) is an even number with prime factor Px.

Still struggle to get to either a proof or a counterexample.

6. Jul 12, 2013

### verty

You seem to be almost there.

$$\frac{x}{w} = \frac{z-1}{y-1}$$

Look at the right hand side and think about the prime decompositions of z-1 and y-1.

7. Jul 12, 2013

### JFGariepy

Thank you very much for helping me like that I really appreciate.

Not being a mathematician, here's what I could come up with, tell me if that makes sense.

From your equation and from knowing that z-1 is even and has prime factor x, and that y-1 is even and has prime factor w, I get:

$$\frac{x}{w} = \frac{2xA}{2wB}$$

where A and B are whatever remaining factors left except 2, x and w in the prime decompositions of the right side of the equation.

Removing the 2 and multiplying each side by w and dividing each side by x, we get

$$\frac{1}{1} = \frac{A}{B}$$

Therefore A = B, that is the only factor that is different in the prime decomposition of z-1 and y-1 is w and x.

I know I must be very close to it, but I still can't see how exactly I will prove something on the original question. This tells me a lot about z-1, but what about z and y ?

8. Jul 12, 2013

### Staff: Mentor

Just test some prime numbers with Px(Py-1) = Pw(Pz-1) and the condition I found, there is an example with very small numbers.

9. Jul 12, 2013

### economicsnerd

These two lines together should do something...

10. Jul 12, 2013

### Staff: Mentor

Please show how you can convert examples for your formulas (like the one I gave) to examples for the original problem.

Those vague suggestions are not useful.

11. Jul 12, 2013

### JFGariepy

I tried a couple and can't find the example. You found some without using 2, which isn't odd ?

3*(13 - 1),5*(11-1)
5*(17 - 1),7*(11-1)
5*(17 - 1),7*(13-1)
3*(17 - 1),7*(13-1)
3*(17 - 1),5*(13-1)
3*(23 - 1),5*(13-1)
3*(29 - 1),5*(19-1)

and many other all gave me inequalities, although I'd love to know what the counterexample is.

------------------------

edit

Ahh my script was somewhat misprogramed.

Here is the list of such sets of primes with primes below 29

3 11 5 7
3 23 11 7
3 29 7 13
5 7 3 11
7 13 3 29
11 7 3 23

This opens up some more mysteries for me as I now have to find some characteristics of those sets of primes, thanks to all of you!

Last edited: Jul 12, 2013
12. Jul 13, 2013

### verty

Try to prove that 3,5,7,11 is the smallest example. You are not yet understanding this relationship.

I think enough help has been given here, let's leave JFGariepy to sort this out.

13. Jul 13, 2013

### JFGariepy

Hum well the fact that 3 5 are the smallest primes to start with and that the other primes have to be at least double of 3 and 5 guarantees that this is the smallest pair.

As for the understanding, I think I do understand that this is the set of primes where the difference between the 1st and 3rd primes is 2 times smaller (or a power of 2 smaller) than the difference between the 2nd and 4th primes. Humm that only holds for small numbers. Will have to work on that.

Last edited: Jul 13, 2013
14. Jul 13, 2013

### Staff: Mentor

3,5,7,11 (=the example I meant) are the four smallest odd primes. How could there be a smaller example?