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Conjecture about primes and the greatest common divisor

5

$begingroup$

Conjecture:

Given $m,ninmathbb N^+$, one odd and one even, there are two
primes $p,q$ such that $|mp-nq|=gcd(m,n)$.

I hope MSE can determine its validity.

From time to time, when testing my growing math packages BigZ and Forthmath, I recognize some patterns which I can't prove or disprove (or even have the ambition to). I post them here with the hope that it will not annoy too much. I hope you can bear with it.

number-theory prime-numbers greatest-common-divisor conjectures

share|cite|improve this question

edited Mar 23 at 19:01

Lehs

asked Mar 23 at 17:04

LehsLehs

6,99731664

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think it is true by the structure of the solution set to Bézout's identity and Dirichlet's theorem on arithmetic progressions. EDIT: But some details need to be taken care of.
  $endgroup$
  – Jeppe Stig Nielsen
  Mar 23 at 17:33
  
  

  
  
  
  


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  2
  

  
  
  
  
  
  

  
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  @JeppeStigNielsen I was thinking the same thing. But the problem is: don't you need to show that two different arithmetic progressions have the property that the $n$th term of each is a prime for some $n$?
  $endgroup$
  – Ashvin Swaminathan
  Mar 23 at 19:36
  

  
  
  
  


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  3
  

  
  
  
  
  
  

  
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  @AshvinSwaminathan Yes, something like that. The two progressions should not "conspire" in a way that they never yield primes simultaneously. I do not know if this is technically hard to establish rigorously. But intuitively, Lehs's conjecture should be true.
  $endgroup$
  – Jeppe Stig Nielsen
  Mar 23 at 19:48
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think this problem is equivalent to an unsolved version of Goldbach's conjecture. Take $m = 1$. Then essentially we're asking the following question: given a linear function $f(x) = a x + b$ with $a,b$ coprime integers, is there some prime $q$ for which $f(q) = p$ is also prime? Some googling suggests this is not known.
  $endgroup$
  – Ashvin Swaminathan
  Mar 24 at 4:02
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
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  @Lehs I think, Ashvin's version is a case of the Bunyakovsky conjecture. I share Ashvin's doubt that this special case is known.
  $endgroup$
  – Peter
  Mar 24 at 14:19
  
  

  
  
  
  

 | 
show 1 more comment

5

$begingroup$

Conjecture:

Given $m,ninmathbb N^+$, one odd and one even, there are two
primes $p,q$ such that $|mp-nq|=gcd(m,n)$.

I hope MSE can determine its validity.

From time to time, when testing my growing math packages BigZ and Forthmath, I recognize some patterns which I can't prove or disprove (or even have the ambition to). I post them here with the hope that it will not annoy too much. I hope you can bear with it.

number-theory prime-numbers greatest-common-divisor conjectures

share|cite|improve this question

edited Mar 23 at 19:01

Lehs

asked Mar 23 at 17:04

LehsLehs

6,99731664

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think it is true by the structure of the solution set to Bézout's identity and Dirichlet's theorem on arithmetic progressions. EDIT: But some details need to be taken care of.
  $endgroup$
  – Jeppe Stig Nielsen
  Mar 23 at 17:33
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @JeppeStigNielsen I was thinking the same thing. But the problem is: don't you need to show that two different arithmetic progressions have the property that the $n$th term of each is a prime for some $n$?
  $endgroup$
  – Ashvin Swaminathan
  Mar 23 at 19:36
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  @AshvinSwaminathan Yes, something like that. The two progressions should not "conspire" in a way that they never yield primes simultaneously. I do not know if this is technically hard to establish rigorously. But intuitively, Lehs's conjecture should be true.
  $endgroup$
  – Jeppe Stig Nielsen
  Mar 23 at 19:48
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think this problem is equivalent to an unsolved version of Goldbach's conjecture. Take $m = 1$. Then essentially we're asking the following question: given a linear function $f(x) = a x + b$ with $a,b$ coprime integers, is there some prime $q$ for which $f(q) = p$ is also prime? Some googling suggests this is not known.
  $endgroup$
  – Ashvin Swaminathan
  Mar 24 at 4:02
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Lehs I think, Ashvin's version is a case of the Bunyakovsky conjecture. I share Ashvin's doubt that this special case is known.
  $endgroup$
  – Peter
  Mar 24 at 14:19
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Conjecture:

Given $m,ninmathbb N^+$, one odd and one even, there are two
primes $p,q$ such that $|mp-nq|=gcd(m,n)$.

I hope MSE can determine its validity.

From time to time, when testing my growing math packages BigZ and Forthmath, I recognize some patterns which I can't prove or disprove (or even have the ambition to). I post them here with the hope that it will not annoy too much. I hope you can bear with it.

number-theory prime-numbers greatest-common-divisor conjectures

share|cite|improve this question

edited Mar 23 at 19:01

Lehs

asked Mar 23 at 17:04

LehsLehs

6,99731664

$endgroup$

share|cite|improve this question

edited Mar 23 at 19:01

Lehs

asked Mar 23 at 17:04

LehsLehs

6,99731664

share|cite|improve this question

edited Mar 23 at 19:01

Lehs

asked Mar 23 at 17:04

LehsLehs

6,99731664

asked Mar 23 at 17:04

LehsLehs

6,99731664

asked Mar 23 at 17:04

LehsLehs

6,99731664