Given $n=2^k p^r q^m$ take a $d\in\Bbb Z$ with $d\mid n$ such that each $a_i$ is in $\{0,\pm1\}$ in $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$

What is the minimum $\gamma$ and the maximum $\tau$ at which

  1. $\forall t\not\equiv t'\bmod(1+ \deg(f_{d,n}(x)))$ $\exists i\in\Bbb Z$ with $0<|i|\leq\gamma$ such that $a_{t+i}\neq a_{t'+i}$ holds

  2. $\exists t\not\equiv t'\bmod(1+ \deg(f_{d,n}(x)))$ such that $\forall 0\leq i\leq\tau$ $a_{t+i}=a_{t'+i}$ holds

    where $t+i,t'+i$ are modulo $1+\deg(f_{d,n}(x))$?

Refer related problem https://mathoverflow.net/questions/290950/could-cyclotomic-polynomials-be-sufficiently-random-to-obtain-certain-combinator.

  • $\begingroup$ The $a_t$ that you ask about in question 1 are the same as the $a_t$ that you implicitly define in question 2? Also, this relates in some way to your other question about these quotients? $\endgroup$ – Gerry Myerson Jan 15 '18 at 18:28
  • $\begingroup$ @GerryMyerson yes $a_t$ are from 2. (I think I need to rephrase) and yes both are related. $\endgroup$ – T.... Jan 15 '18 at 18:31
  • $\begingroup$ Maybe you should link each question to the other? $\endgroup$ – Gerry Myerson Jan 15 '18 at 23:25
  • 1
    $\begingroup$ @GerryMyerson I have cross-linked posts. $\endgroup$ – T.... Jan 16 '18 at 0:47

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