0
$\begingroup$

Let$$2^{\ell-1}<p_1<p_2<\dots<p_t<2^\ell$$$$2^{\ell'-1}<q_1<q_2<\dots<q_m<2^{\ell'}$$be primes on the condition$$\phi(p_i)=2q_1^{a_1}\dots q_m^{a_m}$$($\phi$ is Euler Totient function) where$q_i$ are odd primes and$$a_1,a_2,\dots,a_m\in\mathbb N_{>0}$$ are integers. What is the minimum$\ell,\ell'\in\mathbb N$ for a given$m,t,t'\in\mathbb N$ necessary such that there is an integer$g\in[2,p_1]$ satisfying the property

$$g^{q_1^{b_1}\dots q_m^{b_m}}\equiv1\bmod p_i$$$$1\leq b_1+\dots+b_m\leq t'$$$$\forall i\in\{1,\dots,m\}\mbox{ }0\leq b_i\leq a_i?$$

Is there heuristics or is the problem studied?

askedSep 4 at 20:45
Turbo's user avatar
$\endgroup$
7
  • $\begingroup$What is that function $\lambda$?$\endgroup$CommentedSep 4 at 21:27
  • $\begingroup$en.wikipedia.org/wiki/Carmichael_function it is same as totient for primes.$\endgroup$CommentedSep 4 at 21:29
  • $\begingroup$Renamed to Totient.$\endgroup$CommentedSep 4 at 21:32
  • $\begingroup$Not following. $\varphi(p_i)=p_i-1$ is even, so how could the $q_j$ all be odd?$\endgroup$CommentedSep 4 at 22:10
  • $\begingroup$Good catch has to be multiplied by $2$.$\endgroup$CommentedSep 4 at 22:12

0

You mustlog in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.