3
$\begingroup$

If I have a integer sequence defined as $a_n={n \choose z}\ mod \ y$ for $n,\ x, \ y \in \mathbb Z$, I have found that it is periodic with length: $y\prod_{k=1}^z gcd(e^{\Lambda(k)},y)$, where $\Lambda(k)$ is the von Mangoldt function. (I have not proven this yet, but it seems related to Lucas's and Kummer's theorems.)

Then, if I have a function $f(x,y,z)$ that denotes the frequency of an integer $x$ in one period length of $a_n={n \choose z}\ mod \ y$, it appears that $f(0,2,z)=$ A073138 in OEIS and $f(0,y,2)=$ A034444 in OEIS.

I have this Mathematica code which I have been using to observe the patterns of this function:

f[x_, y_, z_] := Count[Table[Mod[Binomial[n, z], y], {n, 1, y*Product[GCD[E^MangoldtLambda[k], y], {k, 1, z}]}], x]

In addition, $\ f(0,p,z)$ where p is prime seems to have some interesting properties. $\ f(0,11,z)$ demonstrates an interesting repetition:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 22, 31, 40, 49, 58, 67, 76, 85, 94, 103, 112, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 44, 51, 58, 65, 72, 79, 86, 93, 100, . . .}

Here's an image of the above.

Here's an image of the above.

My number theory knowledge is not sufficient to get much further with this problem, and I am currently uncertain how to proceed. Why is this pattern occurring? Can it be proven that this is true? Most importantly to me, how can I express $f(x,y,z)$ in a mathematical way? I have found some papers that talk about the periodicity of $a_n={n \choose z}\ mod \ y$ (like this one, for example), but they are beyond my current understanding. Any help would be appreciated! (This is also my first post, so sorry for any formatting errors.)

This question was previously posted on Math.SE, link.

$\endgroup$
5
  • $\begingroup$ @jeq Sorry, is cross-posting not allowed? It's just that I wasn't getting any answers on the other forum so I thought I'd check here. $\endgroup$ Feb 1, 2018 at 0:51
  • $\begingroup$ @EthanFenlon It's not that it's not allowed, it's that it's 'MO etiquette' to include a link to the cross post in the question. Some people get a little salty about cross posts though, Idk why. $\endgroup$
    – user78249
    Feb 2, 2018 at 2:42
  • 2
    $\begingroup$ I am a little confused - the paper you cite seems to have a complete result, and is also completely elementary (not necessarily in the sense of being easy, though seemingly that too, but in the sense of not using any fancy machinery). That being the case, what exactly are you looking for here? $\endgroup$
    – Igor Rivin
    Feb 2, 2018 at 2:49
  • $\begingroup$ @IgorRivin I am sorry if this result is too elementary or easy to be appropriate for this forum, but I am simply trying to better understand this problem. The paper doesn't seem to touch on the frequency/distribution of numbers in a period length, which is my main area of interest. Or, perhaps it does touch upon this and I simply did not understand it. Either way, I was hoping someone might point me in the right direction. $\endgroup$ Feb 3, 2018 at 20:19
  • $\begingroup$ @EthanFenlon my point is not that the paper is too easy (I haven't read it in detail, so I should not presume), but simply that it seems to contain the answers you seek, and does not use advanced techniques which people here could be asked to explain. It's just a question of reading the paper carefully. $\endgroup$
    – Igor Rivin
    Feb 3, 2018 at 21:32

0