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Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, by wolfram alpha i have got no integer n>1 satisfies the equation $\sigma_y(n)\bmod \sigma_x(n)=0$ at a least it is true from $ n=2$ to $ 5000$ with $y=3$ and $x=2$,According to "Gerhard paseman Idea " we have the following question show the counter example and the problem I challenge is to find integers:$x,p, q ,y$ which presnt the counter example of the precedent equation,

Then i have tried to find these integers $x,p, q ,y$ which satisfy the following conditions but I don't succeed:

$p, q$ are primes, $x$ is a positive integer such that:

$(01)\quad $ $b$ is not a multiple of $x$, but $(1+p^x) $ divides $(1+q^b)$.

$(02)\quad$ $c$ is not a multiple of $x$, but $(1+q^x)$ divides $(1+p^c)$

$(03)\quad$ $\gcd\, (y,x)=1$, where $y={\rm lcm}\,(c,b)$ .

Question. Is there an example of such integers $x,p,q,y$?

Note:I edited the question to show why I posted this question and to make it clear

Thank you for any help .

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closed as unclear what you're asking by Peter LeFanu Lumsdaine, András Bátkai, user1688, user9072, Joonas Ilmavirta Jan 30 '16 at 20:57

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Is your question the following? "Do there exist primes $p,q$ and positive integers $x,b,c$ such that gcd$(x,bc)=1$, $1+p^x$ divides $1+q^b$, $1+q^x$ divides $1+p^c$?" If so, I suggest to formulate it so. $\endgroup$ – Fedor Petrov Jan 27 '16 at 10:57
  • $\begingroup$ This question is a bit unclear. The claim is formulated as though it were a universal statement, or at least an implication: “Let p,q,x be such that (1),(2),(3); then gcd(x,y) = 1.” But then you ask for an example of such integers. Do you mean a counterexample to the implication? Or an example with (1),(2),(3) and with gcd(y,x) = 1? This needs to be clarified, or the question is unanswerable. $\endgroup$ – Peter LeFanu Lumsdaine Jan 27 '16 at 11:09
  • $\begingroup$ Well, I see, you additionally require that $x>1$. $\endgroup$ – Fedor Petrov Jan 27 '16 at 11:51
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This is possible, as expected (usually if there are no obvious reasons why not, the answer in such a problem is yes.)

Try $p=3$, $x=2$. Then we need $10|q^b+1$, $q^2+1|3^c+1$ for some odd $b,c$. First relation is possible for $q=10k-1$, $b=1$. The second holds if the order of $-3$ modulo $q^2+1$ is odd. Computations (with help of Wolframalpha) suggest that $q=59$, $c=435$ work.

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  • $\begingroup$ Interesting. I did not think that c would be large. Are there examples with smaller values of b and c? The motivation is to find $x \lt y$ and primes $p \lt q$ such that both $(1+p^x)(1+ q^x) \mid (1+p^y)(1+q^y)$ and $z = y \bmod x$ hold. Right now, we just care whether $z$ is always 0 or sometimes nonzero, and don't have a specific $z$ value in mind. Is it easy for you to pick a range of pq, and for each p q list those values (y mod x) where one has $(1+p^x)(1+q^x) \mid (1+p^y)(1+q^y)$? Gerhard "Developing Minds Like To Know" Paseman, 2016.01.27 $\endgroup$ – Gerhard Paseman Jan 27 '16 at 16:53
  • $\begingroup$ I do not understand, what is $z$? $\endgroup$ – Fedor Petrov Jan 27 '16 at 17:31
  • $\begingroup$ Z is a number that has been given. At some point, I might care if z is 7. Right now I care only if y is a multiple of x.If there is a pq ,an x, and a y that is y=Nx+7 and the divisibility relation holds (and x is also not 7), I may get excited. If there is no such pq x and y, and you prove that, I will again get excited. But for right now, I would be content with a list of y mod x values in the situation, which I expect will be uniformly distributed with the collection of enough examples. Gerhard "Having Some Understanding Issues Today" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 17:44
  • $\begingroup$ $x=2$, $p=3$, $q=59$, $y=435$, $N=214$, are you really excited? $\endgroup$ – Fedor Petrov Jan 27 '16 at 17:49
  • $\begingroup$ I am already somewhat excited. That's why I commented. Are there examples with smaller y? Gerhard "Looking Forward To More Excitement" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 18:07
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As Fedor Petrov mentions, in the absence of obvious obstructions there will usually be examples.

This question is part of a study by user zeraoulia rafik on the nature of the divisibility relation $\sigma_x(n) \mid \sigma_y(n)$. I have remarked before that the value of $n$ (and not $\tau(n)$ or some more general property of the factorization of $n$) will influence the pairs $(x,y)$ for which the relation holds. In particular, $n=pq$ for $p$ and $q$ distinct primes should provide some examples not of the form $(x,kx)$ for some integer $k \gt 1$.

Fedor Petrov's example of $(p,q)=(3,59)$ shows that we can't always expect $y$ very small given small $p$ and $q$ and $x$. Further $p,q$ pairs with $x=2$ are $(7,29), (11,19),$ and $(17,109)$, for which $y$ can be found using a congruence analysis similar to what Fedor showed. (Assuming no programmer error, these are all the examples with $p$ and $q$ less than $200$ and interesting $x$ and $y$ less than $600$.)

Giving up the idea that $p$ and $q$ would be small, it would be nice to see examples for small values of $x$ other than $2$. Even nicer would be to see some reasoning that would characterize the pairs $(p,q)$ which would participate in $\sigma_x(pq) \mid \sigma_y(pq)$, using an analysis with congruences as above. I suspect the case when $pq$ is replaced by $n$ is rather complex, and will be a challenge to characterize.

Gerhard "Leaves Some Work For Reader" Paseman, 2016.01.27

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