Mihai Prunescu: Symmetries in the Pascal triangle: $p$-adic valuation, sign-reduction modulo $p$ and the last non-zero digit, 431-447

Abstract:

Consider a prime number $p$. Let $v_p$ be the $p$-adic valuation. Let $u_p$ be the sign reduction modulo $p$ defined as $u_p(x) = x$ if $0 \leq x \leq p/2$ and $u_p(x) = p - x$ if $p/2 < x < p$. We say that a triangular numeric pattern $x(a,b)$ with $0 \leq a \leq b \leq n$ has triangular symmetry if it is preserved by the dyhedral group $D_6$. We show the following facts about binomial coefficients:

1. $v_p(\binom{a}{b})$ build a pattern with triangular symmetry for $0 \leq b\leq a \leq p^m - 1$.
2. $u_p(\binom{a}{b} \bmod p)$ build a pattern with triangular symmetry for $0 \leq b\leq a \leq p^m - 1$.
3. $n = 4$ is the only composite number such that $u_n(\binom{a}{b} \bmod n)$ has triangular symmetry for $0 \leq b \leq a \leq n^m - 1$. The fact that $u_4(\binom{a}{b} \bmod 4)$ has triangular symmetry was previously observed by A. Granville.
4. $u_p$ applied to the last non-zero digit of $\binom{a}{b}$ represented in the number system with base $p$ builds a pattern with triangular symmetry for $0 \leq b\leq a \leq p^m - 1$.

Finally, a combined pattern unifies all proven features.

Key Words: Binomial coefficient, $p$-adic valuation, triangular symmetry, Kummer's theorem about carries, Pascal's Triangle modulo $p^k$, automatic $2$-dimensional sequence, Zaphod Beeblebrox.

2010 Mathematics Subject Classification: Primary 11A07; Secondary 05E11, 28A80, 68Q45.