If $ageq 2$, $anmid b$, and $a^n-1mid b^n-1$ for all $ninmathbb{N}$, then $b=1$
Let $a,binmathbb{N}$ be such that $ageq 2$, $anmid b$, and $a^n-1mid b^n-1$ for all $ninmathbb{N}$, then $b=1$.
PS: In fact, if we do not assume that $anmid b$, then the statement should be $b=a^k$ for some $kinmathbb{N}cup{0}$. The above statement can deduce the fact.
elementary-number-theory divisibility
add a comment |
Let $a,binmathbb{N}$ be such that $ageq 2$, $anmid b$, and $a^n-1mid b^n-1$ for all $ninmathbb{N}$, then $b=1$.
PS: In fact, if we do not assume that $anmid b$, then the statement should be $b=a^k$ for some $kinmathbb{N}cup{0}$. The above statement can deduce the fact.
elementary-number-theory divisibility
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23
add a comment |
Let $a,binmathbb{N}$ be such that $ageq 2$, $anmid b$, and $a^n-1mid b^n-1$ for all $ninmathbb{N}$, then $b=1$.
PS: In fact, if we do not assume that $anmid b$, then the statement should be $b=a^k$ for some $kinmathbb{N}cup{0}$. The above statement can deduce the fact.
elementary-number-theory divisibility
Let $a,binmathbb{N}$ be such that $ageq 2$, $anmid b$, and $a^n-1mid b^n-1$ for all $ninmathbb{N}$, then $b=1$.
PS: In fact, if we do not assume that $anmid b$, then the statement should be $b=a^k$ for some $kinmathbb{N}cup{0}$. The above statement can deduce the fact.
elementary-number-theory divisibility
elementary-number-theory divisibility
edited Oct 12 '16 at 5:23
Martin Sleziak
44.7k8115271
44.7k8115271
asked Oct 12 '16 at 4:14
m-agag2016
63448
63448
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23
add a comment |
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23
add a comment |
3 Answers
3
active
oldest
votes
This is a solution here which was found in some paper in AMM. it's a very nice proof. but it's very constructive. For example, the construction of $Q_k(x)$, $p_k$ and $r_{k,n}$, it's not so easy to construct them. Maybe there is some direct method.
add a comment |
Here is my proof from AoPS.
$textbf{Problem 1.}$ Prove that if $b^n-1mid a^n-1$ for all $n$, then $a=b^k$ for some integer $k$.
$textbf{Lemma 1.}$ If $b^n-1mid a^n-1$ for all $n$, then $(b-1)(b^2-1)ldots (b^n-1)mid n!(a-1)(a-b)ldots (a-b^{n-1})$ for every $n$.
$textbf{Proof of Lemma 1.}$ First consider any prime $pmid (b-1)(b^2-1)ldots (b^n-1)$, so there exists minimal $d$ such that $pmid b^d-1$ and it's well-known fact that then $x^d-1equiv (x-1)(x-b)ldots (x-b^{d-1})pmod{p^{nu_p(b^d-1)}}$, so $P(x)=x^d-1$ has exactly $d$ consecutive roots $1, b, b^2,ldots , b^{d-1}$. From the condition $b^d-1mid a^d-1$ we get that $a$ is also the root of polynomial $x^d-1$. So easy to see that there exists only one $0leq d'leq d$, such that $a-b^{d'}equiv 0pmod{p^{nu_p(b^d-1)}}$. From this we get that for all $iequiv d'pmod{d}$, $a-b^iequiv 0pmod{p^{nu_p(b^d-1)}}$. Now from LTE lemma we get that $nu_p(b^{jd}-1)leqnu_p(b^d-1)+nu_p(j)$, for each $j$, so $nu_p(prod_{i=1}^n(b^i-1))leq[n/d]nu_p(b^d-1)+nu_p(n!)leqnu_p(n!(a-1)(a-b)ldots (a-b^{n-1}))$. Combining similar inequalities for every $p$ we get Lemma 1. $Box$
$textbf{Proof of Problem 1.}$ Firstly it's simple exercise to prove that if a prime $p$ is such that $pmid a$, then $pmid b$. So there exists a positive integer $t$, such that $anmid b^{t-1}$, but $amid b^t$. From Lemma 1 we get that $(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}inmathbb{Z}$. But for every $i$ we have that $gcd(a, b^i-1)$, so if we just look on the numerator of the fraction above then we get that $$gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^tmid (2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}$$ and if $anot= b^x$ for every natural $x$, then $$|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|geq gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^t$$ But it's easy to check that $|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|leq (2t)!a^t$ and $gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})geq 2^{frac{t(t-1)}{2}}$ (here we used the fact that $gcd(a, b^i)midgcd(a, b^{i+1})$ and $gcd(a, b^{i-1})not=gcd(a, b^i)$ for $ileq t-1$). So $2^{frac{t(t-1)}{2}}a^tleq (2t)!a^t$. Contradiction. $Box$
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
add a comment |
Reference for a generalization of this theorem and it's proof in French. See this article.
Langlois, Bruno, An arithmetical application of Hadamard’s quotient theorem, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 44, 183-196 (2015). ZBL1389.11046.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f1964788%2fif-a-geq-2-a-nmid-b-and-an-1-mid-bn-1-for-all-n-in-mathbbn-then%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
This is a solution here which was found in some paper in AMM. it's a very nice proof. but it's very constructive. For example, the construction of $Q_k(x)$, $p_k$ and $r_{k,n}$, it's not so easy to construct them. Maybe there is some direct method.
add a comment |
This is a solution here which was found in some paper in AMM. it's a very nice proof. but it's very constructive. For example, the construction of $Q_k(x)$, $p_k$ and $r_{k,n}$, it's not so easy to construct them. Maybe there is some direct method.
add a comment |
This is a solution here which was found in some paper in AMM. it's a very nice proof. but it's very constructive. For example, the construction of $Q_k(x)$, $p_k$ and $r_{k,n}$, it's not so easy to construct them. Maybe there is some direct method.
This is a solution here which was found in some paper in AMM. it's a very nice proof. but it's very constructive. For example, the construction of $Q_k(x)$, $p_k$ and $r_{k,n}$, it's not so easy to construct them. Maybe there is some direct method.
answered Nov 20 '16 at 4:27
m-agag2016
63448
63448
add a comment |
add a comment |
Here is my proof from AoPS.
$textbf{Problem 1.}$ Prove that if $b^n-1mid a^n-1$ for all $n$, then $a=b^k$ for some integer $k$.
$textbf{Lemma 1.}$ If $b^n-1mid a^n-1$ for all $n$, then $(b-1)(b^2-1)ldots (b^n-1)mid n!(a-1)(a-b)ldots (a-b^{n-1})$ for every $n$.
$textbf{Proof of Lemma 1.}$ First consider any prime $pmid (b-1)(b^2-1)ldots (b^n-1)$, so there exists minimal $d$ such that $pmid b^d-1$ and it's well-known fact that then $x^d-1equiv (x-1)(x-b)ldots (x-b^{d-1})pmod{p^{nu_p(b^d-1)}}$, so $P(x)=x^d-1$ has exactly $d$ consecutive roots $1, b, b^2,ldots , b^{d-1}$. From the condition $b^d-1mid a^d-1$ we get that $a$ is also the root of polynomial $x^d-1$. So easy to see that there exists only one $0leq d'leq d$, such that $a-b^{d'}equiv 0pmod{p^{nu_p(b^d-1)}}$. From this we get that for all $iequiv d'pmod{d}$, $a-b^iequiv 0pmod{p^{nu_p(b^d-1)}}$. Now from LTE lemma we get that $nu_p(b^{jd}-1)leqnu_p(b^d-1)+nu_p(j)$, for each $j$, so $nu_p(prod_{i=1}^n(b^i-1))leq[n/d]nu_p(b^d-1)+nu_p(n!)leqnu_p(n!(a-1)(a-b)ldots (a-b^{n-1}))$. Combining similar inequalities for every $p$ we get Lemma 1. $Box$
$textbf{Proof of Problem 1.}$ Firstly it's simple exercise to prove that if a prime $p$ is such that $pmid a$, then $pmid b$. So there exists a positive integer $t$, such that $anmid b^{t-1}$, but $amid b^t$. From Lemma 1 we get that $(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}inmathbb{Z}$. But for every $i$ we have that $gcd(a, b^i-1)$, so if we just look on the numerator of the fraction above then we get that $$gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^tmid (2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}$$ and if $anot= b^x$ for every natural $x$, then $$|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|geq gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^t$$ But it's easy to check that $|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|leq (2t)!a^t$ and $gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})geq 2^{frac{t(t-1)}{2}}$ (here we used the fact that $gcd(a, b^i)midgcd(a, b^{i+1})$ and $gcd(a, b^{i-1})not=gcd(a, b^i)$ for $ileq t-1$). So $2^{frac{t(t-1)}{2}}a^tleq (2t)!a^t$. Contradiction. $Box$
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
add a comment |
Here is my proof from AoPS.
$textbf{Problem 1.}$ Prove that if $b^n-1mid a^n-1$ for all $n$, then $a=b^k$ for some integer $k$.
$textbf{Lemma 1.}$ If $b^n-1mid a^n-1$ for all $n$, then $(b-1)(b^2-1)ldots (b^n-1)mid n!(a-1)(a-b)ldots (a-b^{n-1})$ for every $n$.
$textbf{Proof of Lemma 1.}$ First consider any prime $pmid (b-1)(b^2-1)ldots (b^n-1)$, so there exists minimal $d$ such that $pmid b^d-1$ and it's well-known fact that then $x^d-1equiv (x-1)(x-b)ldots (x-b^{d-1})pmod{p^{nu_p(b^d-1)}}$, so $P(x)=x^d-1$ has exactly $d$ consecutive roots $1, b, b^2,ldots , b^{d-1}$. From the condition $b^d-1mid a^d-1$ we get that $a$ is also the root of polynomial $x^d-1$. So easy to see that there exists only one $0leq d'leq d$, such that $a-b^{d'}equiv 0pmod{p^{nu_p(b^d-1)}}$. From this we get that for all $iequiv d'pmod{d}$, $a-b^iequiv 0pmod{p^{nu_p(b^d-1)}}$. Now from LTE lemma we get that $nu_p(b^{jd}-1)leqnu_p(b^d-1)+nu_p(j)$, for each $j$, so $nu_p(prod_{i=1}^n(b^i-1))leq[n/d]nu_p(b^d-1)+nu_p(n!)leqnu_p(n!(a-1)(a-b)ldots (a-b^{n-1}))$. Combining similar inequalities for every $p$ we get Lemma 1. $Box$
$textbf{Proof of Problem 1.}$ Firstly it's simple exercise to prove that if a prime $p$ is such that $pmid a$, then $pmid b$. So there exists a positive integer $t$, such that $anmid b^{t-1}$, but $amid b^t$. From Lemma 1 we get that $(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}inmathbb{Z}$. But for every $i$ we have that $gcd(a, b^i-1)$, so if we just look on the numerator of the fraction above then we get that $$gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^tmid (2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}$$ and if $anot= b^x$ for every natural $x$, then $$|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|geq gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^t$$ But it's easy to check that $|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|leq (2t)!a^t$ and $gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})geq 2^{frac{t(t-1)}{2}}$ (here we used the fact that $gcd(a, b^i)midgcd(a, b^{i+1})$ and $gcd(a, b^{i-1})not=gcd(a, b^i)$ for $ileq t-1$). So $2^{frac{t(t-1)}{2}}a^tleq (2t)!a^t$. Contradiction. $Box$
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
add a comment |
Here is my proof from AoPS.
$textbf{Problem 1.}$ Prove that if $b^n-1mid a^n-1$ for all $n$, then $a=b^k$ for some integer $k$.
$textbf{Lemma 1.}$ If $b^n-1mid a^n-1$ for all $n$, then $(b-1)(b^2-1)ldots (b^n-1)mid n!(a-1)(a-b)ldots (a-b^{n-1})$ for every $n$.
$textbf{Proof of Lemma 1.}$ First consider any prime $pmid (b-1)(b^2-1)ldots (b^n-1)$, so there exists minimal $d$ such that $pmid b^d-1$ and it's well-known fact that then $x^d-1equiv (x-1)(x-b)ldots (x-b^{d-1})pmod{p^{nu_p(b^d-1)}}$, so $P(x)=x^d-1$ has exactly $d$ consecutive roots $1, b, b^2,ldots , b^{d-1}$. From the condition $b^d-1mid a^d-1$ we get that $a$ is also the root of polynomial $x^d-1$. So easy to see that there exists only one $0leq d'leq d$, such that $a-b^{d'}equiv 0pmod{p^{nu_p(b^d-1)}}$. From this we get that for all $iequiv d'pmod{d}$, $a-b^iequiv 0pmod{p^{nu_p(b^d-1)}}$. Now from LTE lemma we get that $nu_p(b^{jd}-1)leqnu_p(b^d-1)+nu_p(j)$, for each $j$, so $nu_p(prod_{i=1}^n(b^i-1))leq[n/d]nu_p(b^d-1)+nu_p(n!)leqnu_p(n!(a-1)(a-b)ldots (a-b^{n-1}))$. Combining similar inequalities for every $p$ we get Lemma 1. $Box$
$textbf{Proof of Problem 1.}$ Firstly it's simple exercise to prove that if a prime $p$ is such that $pmid a$, then $pmid b$. So there exists a positive integer $t$, such that $anmid b^{t-1}$, but $amid b^t$. From Lemma 1 we get that $(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}inmathbb{Z}$. But for every $i$ we have that $gcd(a, b^i-1)$, so if we just look on the numerator of the fraction above then we get that $$gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^tmid (2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}$$ and if $anot= b^x$ for every natural $x$, then $$|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|geq gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^t$$ But it's easy to check that $|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|leq (2t)!a^t$ and $gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})geq 2^{frac{t(t-1)}{2}}$ (here we used the fact that $gcd(a, b^i)midgcd(a, b^{i+1})$ and $gcd(a, b^{i-1})not=gcd(a, b^i)$ for $ileq t-1$). So $2^{frac{t(t-1)}{2}}a^tleq (2t)!a^t$. Contradiction. $Box$
Here is my proof from AoPS.
$textbf{Problem 1.}$ Prove that if $b^n-1mid a^n-1$ for all $n$, then $a=b^k$ for some integer $k$.
$textbf{Lemma 1.}$ If $b^n-1mid a^n-1$ for all $n$, then $(b-1)(b^2-1)ldots (b^n-1)mid n!(a-1)(a-b)ldots (a-b^{n-1})$ for every $n$.
$textbf{Proof of Lemma 1.}$ First consider any prime $pmid (b-1)(b^2-1)ldots (b^n-1)$, so there exists minimal $d$ such that $pmid b^d-1$ and it's well-known fact that then $x^d-1equiv (x-1)(x-b)ldots (x-b^{d-1})pmod{p^{nu_p(b^d-1)}}$, so $P(x)=x^d-1$ has exactly $d$ consecutive roots $1, b, b^2,ldots , b^{d-1}$. From the condition $b^d-1mid a^d-1$ we get that $a$ is also the root of polynomial $x^d-1$. So easy to see that there exists only one $0leq d'leq d$, such that $a-b^{d'}equiv 0pmod{p^{nu_p(b^d-1)}}$. From this we get that for all $iequiv d'pmod{d}$, $a-b^iequiv 0pmod{p^{nu_p(b^d-1)}}$. Now from LTE lemma we get that $nu_p(b^{jd}-1)leqnu_p(b^d-1)+nu_p(j)$, for each $j$, so $nu_p(prod_{i=1}^n(b^i-1))leq[n/d]nu_p(b^d-1)+nu_p(n!)leqnu_p(n!(a-1)(a-b)ldots (a-b^{n-1}))$. Combining similar inequalities for every $p$ we get Lemma 1. $Box$
$textbf{Proof of Problem 1.}$ Firstly it's simple exercise to prove that if a prime $p$ is such that $pmid a$, then $pmid b$. So there exists a positive integer $t$, such that $anmid b^{t-1}$, but $amid b^t$. From Lemma 1 we get that $(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}inmathbb{Z}$. But for every $i$ we have that $gcd(a, b^i-1)$, so if we just look on the numerator of the fraction above then we get that $$gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^tmid (2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}$$ and if $anot= b^x$ for every natural $x$, then $$|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|geq gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})a^t$$ But it's easy to check that $|(2t)!frac{(a-1)(a-b)ldots (a-b^{t-1})(a-b^t)ldots (a-b^{2t-1})}{(b-1)(b^2-1)ldots (b^t-1)(b^{t+1}-1)ldots (b^{2t}-1)}|leq (2t)!a^t$ and $gcd(a,1)gcd(a, b)ldots gcd(a, b^{t-1})geq 2^{frac{t(t-1)}{2}}$ (here we used the fact that $gcd(a, b^i)midgcd(a, b^{i+1})$ and $gcd(a, b^{i-1})not=gcd(a, b^i)$ for $ileq t-1$). So $2^{frac{t(t-1)}{2}}a^tleq (2t)!a^t$. Contradiction. $Box$
edited Nov 27 '18 at 20:38
answered Nov 27 '18 at 19:58
solver6
5917
5917
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
add a comment |
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that it has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$.
– solver6
Nov 27 '18 at 21:19
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
Can someone find any mistakes in another proof : Let $p$ be a prime number greater than $a$. Consider a polynomial $F(x)=x^p-1$ over $mod{frac{b^p-1}{b-1}}$. One can get that $P(x)$ has $d$ consecutive roots $1, b,ldots , b^{p-1}$. So from the fact $b^p-1mid a^p-1$ we get that $a$ is also root of $F(x)equiv 0pmod{frac{b^p-1}{b-1}}$. Thus, $aequiv b^ipmod{frac{b^p-1}{b-1}}$ for some $ileq p-1$. Finally, if $anot= b^i$, then $b^{p-1}-1geq |a-b^i|geqfrac{b^p-1}{b-1}$. Contradiction. $Box$
– solver6
Nov 27 '18 at 21:26
add a comment |
Reference for a generalization of this theorem and it's proof in French. See this article.
Langlois, Bruno, An arithmetical application of Hadamard’s quotient theorem, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 44, 183-196 (2015). ZBL1389.11046.
add a comment |
Reference for a generalization of this theorem and it's proof in French. See this article.
Langlois, Bruno, An arithmetical application of Hadamard’s quotient theorem, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 44, 183-196 (2015). ZBL1389.11046.
add a comment |
Reference for a generalization of this theorem and it's proof in French. See this article.
Langlois, Bruno, An arithmetical application of Hadamard’s quotient theorem, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 44, 183-196 (2015). ZBL1389.11046.
Reference for a generalization of this theorem and it's proof in French. See this article.
Langlois, Bruno, An arithmetical application of Hadamard’s quotient theorem, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 44, 183-196 (2015). ZBL1389.11046.
answered Jan 3 at 21:03
solver6
5917
5917
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f1964788%2fif-a-geq-2-a-nmid-b-and-an-1-mid-bn-1-for-all-n-in-mathbbn-then%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
Please, try to make the titles of your questions more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Martin Sleziak
Oct 12 '16 at 5:23