# MUIC Math

Mathematics at MUIC

# $p$-adic valuation of Lucas sequences

Aram Tangboonduangjit
Applied Mathematics, Mahidol University International College

16:00 - 17:00 Wednesday, 6th February 2019, Room 1406, MUIC

### Abstract

For relatively prime integers $P$ and $Q$, the Lucas sequence $(U_n)_{n\ge 0} = (U_n(P,Q))_{n\ge 0}$ is defined recursively by $U_0 = 0$, $U_1=1$, and $U_{n} = P\cdot U_{n-1}-Q\cdot U_{n-2}$ for $n\ge 2$. A recent work by Sanna has revealed an astounding formula for the powers of primes in the prime factorization of the Lucas sequence. Sanna's work is a generalization of the work by Lengyel who gave such formula only for the Fibonacci numbers which are the quintessential Lucas sequence with $a=1$ and $b=-1$. In this talk, I will discuss such formula and give applications which are recent joint work with Panraksa.