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

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Discussion Overview

The discussion revolves around the possibility of the equality PxPy + Pw = PwPz + Px being satisfied by four distinct odd prime numbers (Px, Py, Pw, Pz). Participants explore various approaches to understand the conditions under which this equality might hold, as well as the implications of such relationships among prime numbers.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions whether the equality can be satisfied and asks for a proof of impossibility if it cannot be.
  • Another participant suggests an equivalent formulation involving distinct primes p and q such that kp + 1 and kq + 1 are also primes, noting that this is not exactly equivalent to the original problem.
  • Some participants discuss the implications of the equality, rewriting it as Px(Py-1) = Pw(Pz-1) and exploring the divisibility conditions that arise from this formulation.
  • There is mention of specific conditions that must hold for the equality to reflect unique prime factorization, particularly regarding the evenness of (Py-1) and (Pz-1).
  • One participant proposes testing various prime numbers to find examples that satisfy the conditions derived from the equality.
  • Another participant shares a list of sets of primes that meet the conditions, indicating that they have found examples but are still uncertain about the broader implications.
  • Some participants express frustration over the lack of clarity in converting examples back to the original problem and seek further guidance.
  • There is a discussion about the smallest example of primes that could satisfy the conditions, with some participants asserting that 3, 5, 7, and 11 are indeed the smallest odd primes.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the equality can be satisfied or how to definitively prove its impossibility. Multiple competing views and approaches are presented, with ongoing exploration of the problem.

Contextual Notes

Some participants note the importance of the primes being distinct and odd, which adds complexity to the problem. There are unresolved mathematical steps and assumptions regarding the relationships between the primes that remain open for further exploration.

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

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.
 
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.
 
verty said:
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.

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 ?
 
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.
 
mfb said:
With k=2, there are many such primes p,q. Examples: p=3, q=5.

JFGariepy said:
Px(Py-1) = Pw(Pz-1)

These two lines together should do something...
 
  • #10
economicsnerd said:
These two lines together should do something...
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
mfb said:
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.

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:
  • #12
JFGariepy said:
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!

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
verty said:
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.

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:
  • #14
3,5,7,11 (=the example I meant) are the four smallest odd primes. How could there be a smaller example?

verty said:
I think enough help has been given here, let's leave JFGariepy to sort this out.
He did that already :confused:.
 
  • #15
JFGariepy said:
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

I see the word "odd" now. I missed it before because the title does not use it and it even says "4 distinct prime numbers" in the quote above. If we are talking about odd primes then obviously 3,5,7,11 is the smallest case.
 

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