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 .


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

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  • 3
    $\begingroup$ 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

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.

  • $\begingroup$ 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

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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