Pascal's triangle shows the factors of a row n.

Question

For a given row n, the value of an element in that row divided by n will produce a value whose denominator is a factor of n.

As shown here:

https://i.stack.imgur.com/dMG2E.jpg

https://i.stack.imgur.com/zth2k.jpg

What I am wondering is if this is a known fact and if it could potentially be used as an efficient algorithm for integer factorization.

Answer 1

It's obvious that every denominator is a factor of $n$, because you start with a denominator $n$ and then cancel common factors (if any).

It's not true that every factor of $n$ occurs as a possible denominator. This is actually in your linked image - if you look carefully at the row for $n=12$, you will see that the denominator is never $6$. The fractions are $$\frac{1}{12},\quad \frac{1}{1},\quad \frac{11}{2},\quad \frac{55}{3},\quad \frac{165}{4},\quad \frac{66}{1},\quad \frac{77}{1}$$ and the same again in reverse.