Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, then I would like to find some examples of integers $n$ and $m$ such that $q^4$ divides $n!$.
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$\begingroup$ @barak, your set of primes may be empty; actually, it always is empty unless $\ m=r_n.\ $ For instance, Tina's $\ q\ $ for $\ (m\ n)\ =\ (8\ \;10)\ $ is $\ q=5.\ $ Thus it's not true that $\ q\ge m$. $\endgroup$– Włodzimierz HolsztyńskiCommented Jun 21, 2014 at 8:52
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$\begingroup$ Except that I missed the condition $\ q\ge 17$. $\endgroup$– Włodzimierz HolsztyńskiCommented Jun 21, 2014 at 9:01
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1$\begingroup$ The condition $q\geq 17$ is important to me. I think in this case we should have $n-m\geq 15$. $\endgroup$– TinaCommented Jun 21, 2014 at 9:03
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$\begingroup$ @Stefan Kohl: Could you please give me some examples for the case that $q^4$ divides $n!$? $\endgroup$– TinaCommented Jun 21, 2014 at 10:10
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3$\begingroup$ @Tina: You are welcome! -- Though to avoid a mismatch between question and answer, I have rolled back your question to the version I have answered. If you are interested in an answer to the new question (where you require divisibility of $n!/m!$ by $q^s$), I suggest you to ask a new question. $\endgroup$– Stefan Kohl ♦Commented Jun 23, 2014 at 19:22
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Show 5 more comments
1 Answer
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1
Given a positive integer $n$, let $P(n)$ denote the largest prime factor of $n$. What you are looking for are integers $n$ such that $q := \max\{P(n-15),P(n-14),\dots,P(n)\} < Cn$ for some constant $C \in ]0,1[$ depending on whether you require $q^2$ to divide $n!$ (as in the first version of your question) or $q^4$ like now or some higher power. However by
John G. Kemeny: Largest prime factor, Journal of Pure and Applied Algebra 89(1993) Issues 1-2, 181-186
the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$.
A list of examples $(n,q,s)$ where $n \leq 10000$ is as follows:
[ [ 539, 269, 2 ], [ 540, 269, 2 ], [ 903, 449, 2 ], [ 904, 449, 2 ],
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[ 1967, 983, 2 ], [ 1968, 983, 2 ], [ 1969, 983, 2 ], [ 1970, 983, 2 ],
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answered Jun 21, 2014 at 20:51
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$\begingroup$ Dear Stefan Kohl, thank you so much for your answer. Actually, I am working on alternating groups and in all your examples $q^2\nmid n!/m!$. My proofs works for these cases. So, I have to edit my question again. By the way, if $n=1342$ and $m=1327$, then $q=443$ and not $q=269$. $\endgroup$– TinaCommented Jun 23, 2014 at 11:32