Wikipedia:Reference desk/Archives/Mathematics/2023 December 13

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 13

New property of Pascal's triangle

(Please read the post in this desk from early December before commenting here.)

If p is a prime number and n is the product of a power of p and an integer less than p, the nth row of Pascal's triangle has no multiples of p. The 15th row is:

1,14,91,364,1001,2002,3003,3432,3003,2002,1001,364,91,14,1

15 is the product of 5 (a power of 5) and 3 (a number less than 5) so the above rule suggests that the row should have no multiples of 5. 2145 is a multiple of 5 and can be ruled out finally. Georgia guy (talk) 16:30, 13 December 2023 (UTC)

It's interesting to note that this is because, if we write $n=ap^{k}-1$ for $1\leq a<p$, which in base $p$ is $(a-1)p^{k}+(p-1)p^{k-1}+\ldots +(p-1)p+(p-1)$, the only way for $p$ to divide $n \choose m$ would be if any digit of $m$ was greater than its corresponding digit in $n$, which isn't possible (there are no base $p$ digits greater than $p-1$, and having a first digit greater than or equal to $a$ makes $m$ greater than $n$ itself.) I'll try to simulate some more values of the triangle later. GalacticShoe (talk) 14:58, 14 December 2023 (UTC)

GalacticShoe, can you now do the same thing that I asked you to do earlier with finding when the triangle deviates from Pascal's triangle, only this time with the new restriction I gave here?? Georgia guy (talk) 16:13, 14 December 2023 (UTC)

Why do these threads keep referring to Pascal's triangle when it isn't? It is just a number triangle with weird rules the OP has come up with. Very confusing. -- SGBailey (talk) 13:33, 15 December 2023 (UTC)

Rather than Pascal's triangle, these triangles are defined by rules that Pascal's triangle satisfies, aside from the explicit additivity condition. I presume Georgia guy is looking to find conditions that keep the triangle closest to Pascal's triangle for the longest time without deviating (possibly even completely matching Pascal's triangle with the right conditions.) GalacticShoe (talk) 14:44, 15 December 2023 (UTC)

GalacticShoe, can you find where this new triangle deviates from Pascal's triangle?? Please try your best. Georgia guy (talk) 14:52, 15 December 2023 (UTC)

Now, let me try. Please correct any mistakes:

• 15th row: 1-14-91-364-1001-2002-3003-3432-3003-2002-1001-364-91-14-1
• 16th row: 1-15-105-455-1365-3003-5005-6435-6435-5005-3003-1365-455-105-15-1
• 17th row: 1-16-120-560-1820-3276(Pascal's triangle has 4368)-8008-11440-12870-11440-8008-3276(once again, as opposed to 4368)-1820-560-120-16-1. Any corrections?? Georgia guy (talk) 15:16, 15 December 2023 (UTC)

  I've just simulated values for the triangle, and your rows are correct. This time, it breaks down in the 18th row, going $1,17,136,680,2380,4004,30940$ before running into trouble for the value that is between $8008$ and $11440$ of the previous row, presumably in part due to the increasing nature of each row. GalacticShoe (talk) 01:23, 16 December 2023 (UTC)