Mixing
Conjecture: $n>2$ is prime iff $sum_{k=1}^{n-1}left(3^k-2right)^{n-1} ;equiv; n cdot 2^{n-1}-1...
$begingroup$
This question is closely related to: Conjectured primality test
Can you provide a proof or a counterexample for the following claim :
Conjecture. Let $n$ be a natural number greater than $2$. Then $n$ is prime if and only if
$$sum_{k=1}^{n-1}left(3^k-2right)^{n-1} ;equiv;
n cdot 2^{n-1}-1 pmod{frac{3^n-1}{2}}$$
You can run this test here.
I was searching for a counterexample using the following two PARI/GP programs :
CE1(n1,n2)=
{
forcomposite(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if((Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
CE2(n1,n2)=
{
forprime(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if(!(Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
REMARK
More generally we can formulate the following criterion :
Let $b$ , $a$ and $n$ be a natural numbers, $b>ageq 1$, $n>2$ and $n notin {4,8,9}$. Then $n$ is prime if and only if
$$displaystylesum_{k=1}^{n}left(b^kpm aright)^{n-1} equiv n cdot a^{n-1} pmod{frac{b^n-1}{b-1}}$$
elementary-number-theory prime-numbers examples-counterexamples primality-test
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
$endgroup$
add a comment |
$begingroup$
This question is closely related to: Conjectured primality test
Can you provide a proof or a counterexample for the following claim :
Conjecture. Let $n$ be a natural number greater than $2$. Then $n$ is prime if and only if
$$sum_{k=1}^{n-1}left(3^k-2right)^{n-1} ;equiv;
n cdot 2^{n-1}-1 pmod{frac{3^n-1}{2}}$$
You can run this test here.
I was searching for a counterexample using the following two PARI/GP programs :
CE1(n1,n2)=
{
forcomposite(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if((Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
CE2(n1,n2)=
{
forprime(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if(!(Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
REMARK
More generally we can formulate the following criterion :
Let $b$ , $a$ and $n$ be a natural numbers, $b>ageq 1$, $n>2$ and $n notin {4,8,9}$. Then $n$ is prime if and only if
$$displaystylesum_{k=1}^{n}left(b^kpm aright)^{n-1} equiv n cdot a^{n-1} pmod{frac{b^n-1}{b-1}}$$
elementary-number-theory prime-numbers examples-counterexamples primality-test
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
$endgroup$
2
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
1
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54
add a comment |
$begingroup$
This question is closely related to: Conjectured primality test
Can you provide a proof or a counterexample for the following claim :
Conjecture. Let $n$ be a natural number greater than $2$. Then $n$ is prime if and only if
$$sum_{k=1}^{n-1}left(3^k-2right)^{n-1} ;equiv;
n cdot 2^{n-1}-1 pmod{frac{3^n-1}{2}}$$
You can run this test here.
I was searching for a counterexample using the following two PARI/GP programs :
CE1(n1,n2)=
{
forcomposite(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if((Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
CE2(n1,n2)=
{
forprime(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if(!(Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
REMARK
More generally we can formulate the following criterion :
Let $b$ , $a$ and $n$ be a natural numbers, $b>ageq 1$, $n>2$ and $n notin {4,8,9}$. Then $n$ is prime if and only if
$$displaystylesum_{k=1}^{n}left(b^kpm aright)^{n-1} equiv n cdot a^{n-1} pmod{frac{b^n-1}{b-1}}$$
elementary-number-theory prime-numbers examples-counterexamples primality-test
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
$endgroup$
This question is closely related to: Conjectured primality test
Can you provide a proof or a counterexample for the following claim :
Conjecture. Let $n$ be a natural number greater than $2$. Then $n$ is prime if and only if
$$sum_{k=1}^{n-1}left(3^k-2right)^{n-1} ;equiv;
n cdot 2^{n-1}-1 pmod{frac{3^n-1}{2}}$$
You can run this test here.
I was searching for a counterexample using the following two PARI/GP programs :
CE1(n1,n2)=
{
forcomposite(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if((Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
CE2(n1,n2)=
{
forprime(n=n1,n2,
s=sum(k=1,n-1,lift(Mod(3^k-2,(3^n-1)/2)^(n-1)));
if(!(Mod(s,(3^n-1)/2)==n*2^(n-1)-1),print("n="n)))
}
REMARK
More generally we can formulate the following criterion :
Let $b$ , $a$ and $n$ be a natural numbers, $b>ageq 1$, $n>2$ and $n notin {4,8,9}$. Then $n$ is prime if and only if
$$displaystylesum_{k=1}^{n}left(b^kpm aright)^{n-1} equiv n cdot a^{n-1} pmod{frac{b^n-1}{b-1}}$$
elementary-number-theory prime-numbers examples-counterexamples primality-test
elementary-number-theory prime-numbers examples-counterexamples primality-test
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
edited Feb 4 at 8:07
Peđa Terzić
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
asked Feb 2 at 9:43
Peđa TerzićPeđa Terzić
7,66922674
7,66922674
2
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
1
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54
add a comment |
2
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
1
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54
2
2
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
1
1
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3097150%2fconjecture-n2-is-prime-iff-sum-k-1n-1-left3k-2-rightn-1-equi%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3097150%2fconjecture-n2-is-prime-iff-sum-k-1n-1-left3k-2-rightn-1-equi%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
$begingroup$
To how high a value of $n$ is it known to hold?
$endgroup$
– Oscar Lanzi
Feb 2 at 10:50
1
$begingroup$
@OscarLanzi For all $n$ up to $5000$.
$endgroup$
– Peđa Terzić
Feb 2 at 12:54