'Pascal's Triangle can be defined in one of two ways (both are equivalent). Sum Definition: The outer entries are 1 by definition, $P_{i,1} = 1$ and $P_{i,i}=1$, and the other entries $P_{i,j}$ are defined to be the sum of the two entries above it $P_{i,j} = P_{i-1,j-1} + P_{i-1,j}$ Binomial Coefficient Definition: The $i,j$ entry is defined to be the Binomial Coefficient $P_{i,j}=\binom{i}{j}$ . Problems: 1. Prove that $P_{i,j}$ defined using the sum definition satisfies $P_{i,j} = \binom{i}{j}$ 2. Prove that $P_{i,j}$ defined as a binomial coefficient $P_{i,j}=\binom{i}{j}$ satisfies $P_{i,j} = P_{i-1,j-1} + P_{i-1,j}$ 3. Prove (using either definition) that the sum of each row is $2^{i}$ 4. Prove that if $p$ is a prime number, then on the $p$-th row all the numbers are divisible by $p$ . (This fact is used in one of the proofs of [[Fermat's Little Theorem]]) This last property can be used to show that if you look at Pascal's triangle mod p, then it will have a row of all 0s at the p-th row. But if it has a row of all 0's, you can see that we are essentially starting over from nothing. So the pattern will be periodic in a certain way. This leads to a nice fractal structure: see these plots below (which were generated from https://colab.research.google.com/drive/1MJbvssjMHserGoWcbju8u8aJLC1wvBYa?usp=sharing ) ![[p2.png]] ![[p3.png]] ![[p5.png]] ![[p7.png]] ![[p11.png]] ![[p13.png]]