Dear all,

**Talk Announcement:**

**Title:**On Fractal Subsets of Pascal's "Pyramid" and the Number of Multinomial Coefficients Congruent to a Given Residue Modulo a Prime

**Speaker:**David Bradley (University of Maine, USA)

**When:**Thursday, Thursday Dec 7 23, 2023 - 6:30 PM (IST) (8AM EST/ 2PM (CET))

**Where**: Zoom: Ask the organisers for a link

**Abstract.**

*p*rows of Pascal’s "pyramid" which are congruent to a given nonzero residue

*r*modulo the prime

*p*. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes

*p*, the number of

*coefficients in Pascal's*

**binomial***which are congruent to*

**triangle***r*mod

*p*for

*r*= 1, 1 <

*r*<

*p*−1, and

*r*=

*p*−1 is well approximated by the respective linear functions of

*p*given

*p*,

*p*/2, and

*p*. In particular, for large primes

*p*there are approximately six times as many occurrences of the residue 1 in the first

*p*rows of Pascal’s triangle reduced modulo

*p*than there are of any other residue

*r*in the range 1 <

*r*<

*p*− 1, and three times as many as

*r*=

*p*− 1. On the other hand, if we let the nonnegative integer

*k*vary while keeping the prime

*p*fixed, and look at the relative frequency of various residue classes that occur in the first

*p*

*k*rows, the seemingly substantial differences in frequency between

*r*= 1, 1 <

*r*<

*p*−1, and

*r = p*−1 when

*k*= 1 are increasingly dissipated as

*k*grows without bound. We show that in the limit as

*k*tends to infinity, all nonzero residues are equally represented with asymptotic proportion 1/(

*p*− 1).