Analog of Cramer's conjecture for primes in a residue class
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Let $q$ and $r$ be fixed coprime positive integers,
$$
1 le r < q, qquad gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy
$$
p equiv p' equiv r ({rm mod} q), tag{1}
$$
and no other primes between $p$ and $p'$ satisfy $(1)$.
Then we have the following
Naive generalization of Cramer's conjecture to primes in residue class $r$ mod $q$:
$$
p'-p ~<~ varphi(q),(ln p')^2. tag{2}
$$
(PrimePuzzles Conjecture 77, A. Kourbatov, 2016).
See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $varphi(q)$ constant.
Here, as usual, $varphi(q)$ denotes Euler's totient function.
Note: In the inequality $(2)$ we take the logarithm of the prime $p'$ at the larger end of the "gap".
Very few counterexamples to $(2)$ are known; see Appendix 7.4 in arXiv:1610.03340. Definitely no counterexamples for $q=2, p<4cdot10^{18}$; also none for $1le r < q le 1000$, $ p<10^{10}$.
This conjecture (mostly in a less-naive "almost always" form) is mentioned in the following OEIS sequences listing maximal (record) gaps between primes of the form $p=qk+r$, $ gcd(q,r)=1$:
A084162,
A268799,
A268925,
A268928,
A268984,
A269234,
A269238,
A269261,
A269420,
A269424,
A269513,
A269519.
Question 1: Find a counterexample to conjecture $(2)$.
Question 2: Find a counterexample to $(2)$, with prime $q$ and prime $r$.
Question 3: Find a counterexample to $(2)$, with
$$
{p'-p over varphi(q)(ln p')^2} > 1.1 tag{3}
$$
(A.Granville predicts that such counterexamples exist even for $q=2$, with the above ratio greater than $1.12$ -- more precisely, Granville expects that the ratio should exceed or come close to $2e^{-gamma}$).
Hint: Counterexamples are very rare. To find one, you will likely need to write a program and run it long enough. Good luck!
number-theory prime-numbers conjectures prime-gaps
asked May 6 '17 at 21:26
Alex
4,2251528
|
show 2 more comments
up vote
6
down vote
favorite
Let $q$ and $r$ be fixed coprime positive integers,
$$
1 le r < q, qquad gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy
$$
p equiv p' equiv r ({rm mod} q), tag{1}
$$
and no other primes between $p$ and $p'$ satisfy $(1)$.
Then we have the following
Naive generalization of Cramer's conjecture to primes in residue class $r$ mod $q$:
$$
p'-p ~<~ varphi(q),(ln p')^2. tag{2}
$$
(PrimePuzzles Conjecture 77, A. Kourbatov, 2016).
See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $varphi(q)$ constant.
Here, as usual, $varphi(q)$ denotes Euler's totient function.
Note: In the inequality $(2)$ we take the logarithm of the prime $p'$ at the larger end of the "gap".
Very few counterexamples to $(2)$ are known; see Appendix 7.4 in arXiv:1610.03340. Definitely no counterexamples for $q=2, p<4cdot10^{18}$; also none for $1le r < q le 1000$, $ p<10^{10}$.
This conjecture (mostly in a less-naive "almost always" form) is mentioned in the following OEIS sequences listing maximal (record) gaps between primes of the form $p=qk+r$, $ gcd(q,r)=1$:
A084162,
A268799,
A268925,
A268928,
A268984,
A269234,
A269238,
A269261,
A269420,
A269424,
A269513,
A269519.
Question 1: Find a counterexample to conjecture $(2)$.
Question 2: Find a counterexample to $(2)$, with prime $q$ and prime $r$.
Question 3: Find a counterexample to $(2)$, with
$$
{p'-p over varphi(q)(ln p')^2} > 1.1 tag{3}
$$
(A.Granville predicts that such counterexamples exist even for $q=2$, with the above ratio greater than $1.12$ -- more precisely, Granville expects that the ratio should exceed or come close to $2e^{-gamma}$).
Hint: Counterexamples are very rare. To find one, you will likely need to write a program and run it long enough. Good luck!
number-theory prime-numbers conjectures prime-gaps
asked May 6 '17 at 21:26
Alex
4,2251528
I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13
|
show 2 more comments
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Let $q$ and $r$ be fixed coprime positive integers,
$$
1 le r < q, qquad gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy
$$
p equiv p' equiv r ({rm mod} q), tag{1}
$$
and no other primes between $p$ and $p'$ satisfy $(1)$.
Then we have the following
Naive generalization of Cramer's conjecture to primes in residue class $r$ mod $q$:
$$
p'-p ~<~ varphi(q),(ln p')^2. tag{2}
$$
(PrimePuzzles Conjecture 77, A. Kourbatov, 2016).
See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $varphi(q)$ constant.
Here, as usual, $varphi(q)$ denotes Euler's totient function.
Note: In the inequality $(2)$ we take the logarithm of the prime $p'$ at the larger end of the "gap".
Very few counterexamples to $(2)$ are known; see Appendix 7.4 in arXiv:1610.03340. Definitely no counterexamples for $q=2, p<4cdot10^{18}$; also none for $1le r < q le 1000$, $ p<10^{10}$.
This conjecture (mostly in a less-naive "almost always" form) is mentioned in the following OEIS sequences listing maximal (record) gaps between primes of the form $p=qk+r$, $ gcd(q,r)=1$:
A084162,
A268799,
A268925,
A268928,
A268984,
A269234,
A269238,
A269261,
A269420,
A269424,
A269513,
A269519.
Question 1: Find a counterexample to conjecture $(2)$.
Question 2: Find a counterexample to $(2)$, with prime $q$ and prime $r$.
Question 3: Find a counterexample to $(2)$, with
$$
{p'-p over varphi(q)(ln p')^2} > 1.1 tag{3}
$$
(A.Granville predicts that such counterexamples exist even for $q=2$, with the above ratio greater than $1.12$ -- more precisely, Granville expects that the ratio should exceed or come close to $2e^{-gamma}$).
Hint: Counterexamples are very rare. To find one, you will likely need to write a program and run it long enough. Good luck!
number-theory prime-numbers conjectures prime-gaps
asked May 6 '17 at 21:26
Alex
4,2251528
Let $q$ and $r$ be fixed coprime positive integers,
$$
1 le r < q, qquad gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy
$$
p equiv p' equiv r ({rm mod} q), tag{1}
$$
and no other primes between $p$ and $p'$ satisfy $(1)$.
Then we have the following
Naive generalization of Cramer's conjecture to primes in residue class $r$ mod $q$:
$$
p'-p ~<~ varphi(q),(ln p')^2. tag{2}
$$
(PrimePuzzles Conjecture 77, A. Kourbatov, 2016).
See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $varphi(q)$ constant.
Here, as usual, $varphi(q)$ denotes Euler's totient function.
Note: In the inequality $(2)$ we take the logarithm of the prime $p'$ at the larger end of the "gap".
Very few counterexamples to $(2)$ are known; see Appendix 7.4 in arXiv:1610.03340. Definitely no counterexamples for $q=2, p<4cdot10^{18}$; also none for $1le r < q le 1000$, $ p<10^{10}$.
This conjecture (mostly in a less-naive "almost always" form) is mentioned in the following OEIS sequences listing maximal (record) gaps between primes of the form $p=qk+r$, $ gcd(q,r)=1$:
A084162,
A268799,
A268925,
A268928,
A268984,
A269234,
A269238,
A269261,
A269420,
A269424,
A269513,
A269519.
Question 1: Find a counterexample to conjecture $(2)$.
Question 2: Find a counterexample to $(2)$, with prime $q$ and prime $r$.
Question 3: Find a counterexample to $(2)$, with
$$
{p'-p over varphi(q)(ln p')^2} > 1.1 tag{3}
$$
(A.Granville predicts that such counterexamples exist even for $q=2$, with the above ratio greater than $1.12$ -- more precisely, Granville expects that the ratio should exceed or come close to $2e^{-gamma}$).
Hint: Counterexamples are very rare. To find one, you will likely need to write a program and run it long enough. Good luck!
number-theory prime-numbers conjectures prime-gaps
number-theory prime-numbers conjectures prime-gaps
asked May 6 '17 at 21:26
Alex
4,2251528
asked May 6 '17 at 21:26
Alex
4,2251528
edited Oct 11 '17 at 10:25
asked May 6 '17 at 21:26
Alex
4,2251528
asked May 6 '17 at 21:26
Alex
4,2251528
asked May 6 '17 at 21:26
Alex
4,2251528
4,2251528
I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13
|
show 2 more comments
I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13
I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13
|
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
0
down vote
Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$.
It is not difficult to check that
$$
p equiv p' equiv 341 ({rm mod} 1605), tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > varphi(q) (log3624431)^2 = 193434.64ldots$$
answered Sep 19 '17 at 8:09
Alex
4,2251528
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$.
It is not difficult to check that
$$
p equiv p' equiv 341 ({rm mod} 1605), tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > varphi(q) (log3624431)^2 = 193434.64ldots$$
answered Sep 19 '17 at 8:09
Alex
4,2251528
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
add a comment |
up vote
0
down vote
Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$.
It is not difficult to check that
$$
p equiv p' equiv 341 ({rm mod} 1605), tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > varphi(q) (log3624431)^2 = 193434.64ldots$$
answered Sep 19 '17 at 8:09
Alex
4,2251528
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
add a comment |
up vote
0
down vote
up vote
0
down vote
Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$.
It is not difficult to check that
$$
p equiv p' equiv 341 ({rm mod} 1605), tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > varphi(q) (log3624431)^2 = 193434.64ldots$$
answered Sep 19 '17 at 8:09
Alex
4,2251528
Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$.
It is not difficult to check that
$$
p equiv p' equiv 341 ({rm mod} 1605), tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > varphi(q) (log3624431)^2 = 193434.64ldots$$
answered Sep 19 '17 at 8:09
Alex
4,2251528
answered Sep 19 '17 at 8:09
Alex
4,2251528
answered Sep 19 '17 at 8:09
Alex
4,2251528
answered Sep 19 '17 at 8:09
Alex
4,2251528
4,2251528
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
add a comment |
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
This only answers question 1. No counterexamples answering questions 2, 3 thus far.
– Alex
Oct 11 '17 at 10:26
add a comment |
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I do expect a counterexample. I only think that such counterexamples are very rare. Thank you!
– Alex
May 8 '17 at 20:06
No you don't except someone will find a counter-example if you already made some programs. That's why you need to make it clear...
– reuns
May 8 '17 at 20:13
Is that $log log p' $ or $(log p')^2 $?
– daniel
May 15 '17 at 15:47
That's $(log p')^2$.
– Alex
May 15 '17 at 16:42
Did you try discussing the random models underlying those things ? (in particular what is Maier's heuristic under which $log^2(p)$ should be replaced by $log^{2+epsilon}(p)$)
– reuns
Nov 14 '17 at 15:13