Date of this Version
Warneke, Z. (2019). Pascal's Triangle Modulo n and Its Applications to Efficient Computation of Binomial Coefficients. Undergraduate Honors Thesis. University of Nebraska-Lincoln.
In this thesis, Pascal's Triangle modulo n will be explored for n prime and n a prime power. Using the results from the case when n is prime, a novel proof of Lucas' Theorem is given. Additionally, using both the results from the exploration of Pascal's Triangle here, as well as previous results, an efficient algorithm for computation of binomial coefficients modulo n (a choose b mod n) is described, and its time complexity is analyzed and compared to naive methods. In particular, the efficient algorithm runs in O(n log(a)) time (as opposed to the naive method's O(a) time), which is highly preferable when n << a. As a supplement, a program to generate Pascal's Triangle modulo n is provided, as well as an implementation of the efficient binomial coefficient computation algorithm described earlier (along with practical time comparisons).