Kanazawa University University of the the Philippines Baguio
Joint Seminar in Mathematics: Algebra and Number Theory
August 1, 2022, via Zoom

12:55 - 13:00 JST / 11:55 - 12:00 PSTOpening

13:00 - 15:00 JST / 12:00 - 14:00 PST

Session (Moderator: Julius Fergy Rabago)

S. = Speaker· T. = Title· A. = Abstract

13:00 - 13:50 JST / 12:00 - 12:50 PSTFirst Talk

S.

Manabu Oura
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University,
oura@se.kanazawa-u.ac.jp

T.

Some topics around association schemesA.
I review what I learned from my teachers and friends. In particular we discuss some of our resultsaround association schemes.
13:55 - 14:15 JST / 12:55 - 13:15 PSTS.

Second Talk

Gina May R. Natividad1,∗ , Edna N. Gueco2 and Dennis I. Merino3
1,2
Department of Mathematics and Computer Science, University of the Philippines Baguio
1
Mathematics Department, Pangasinan State University, Lingayen, Pangasinan
3
Department of Mathematics, Southeastern Louisiana University
1
grnatividad@up.edu.ph, 2 engueco@up.edu.ph, 3 dmerino@selu.edu

T.

J-Householder matrices over Zn"#0 IA.
Let J =. We say that A ∈ M2k (Zn ) is J-orthogonal or symplectic if J −1 A> J = A−1 and A−I 0is J-symmetric if J −1 A> J = A. A matrix H ∈ M2k (Zn ) is said to be a J-Householder if H is symplectic andrank (H − I) = 1. We show that J-Householder matrices have the form I − αxx> J, for any scalar α and2k>0 6= x ∈ Z2kn . Let u ∈ Zn with uu 6= 0 be given. The J-Householder matrix corresponding to u is given by
Hu = I − uu> J.
We determine forms of the power, inverse, transpose, and the Jordan canonical form of J-Householder
matrices corresponding to a vector in M2k (Zn ). We also present properties of J-Householder matrices that
hold in M2k (Zn ) when n is an odd prime.
References:[1] K.L. de la Rosa, D.I. Merino, and A.T. Paras, The J-Householder matrices, Linear Algebra and Its Applications 436 (2012) 1189-1194.
[2] R.A. Horn, and C.R. Johnson, Matrix Analysis. Cambridge University Press, New York, (1985).

14:15 - 14:35 JST / 13:15 - 13:35 PST

Third Talk

S.

Richard J. Taclay1,∗ , Jerico B. Bacani2
1,2
Department of Mathematics and Computer Science, University of the Philippines Baguio
1
Nueva Vizcaya State University-Bayombong Campus, Nueva Vizcaya
1
rjtaclay@up.edu.ph, 2 jbbacani@up.edu.phT.

On the quartic Diophantine equation 2a px4 + y 4 = z 4

A.
In this paper, we consider the quartic Diophantine equation 2a px4 + y 4 = z 4 , where p is a prime
number, x, y, z ∈ Z and a ∈ Z+ . We show the impossibility of primitive nonzero integer solutions of theequation under consideration if p ≡ 3 (mod 4), p 6≡ 1 (mod 16), p ≡ 3, 4 (mod 5), p ≡ 7, 8, 11 (mod 13),and p ≡ 4, 5, 6, 9, 13, 22, 28 (mod 29), given that a = 2k + 3, k ∈ N0 . Also, we provide parametric solutions
for the case p = 4a−3 I 8 + J 8 , where I, J ∈ Z. Moreover, if p is the nth Fermat prime, the quartic diophantine
equation has a solution for some a = 2n−1 + 3, n ∈ Z+ .
14:35 - 14:55 JST / 13:35 - 13:55 PST

Last Talk

S.

William S. Gayo, Jr.
College of Arts and Sciences, Don Mariano Marcos Memorial State University,
North La Union Campus, wgayo@dmmmsu.edu.phT.

The search for integer solutions of equations of the form ax +by =z 2 : A Diophantine Analysis

A.
Diophantine analysis is one of the major trends in Number Theory researches these days. Its main
goal is to seek the solutions of Diopantine equations like the equation ax + by = z 2 , which has been studied
extensively for the past decades. In this talk, we will present elementary methods and concepts in solving
this Diophantine equation. Moreover, solutions to specific forms of this equation will be tackled.
14:55 - 15:00 JST / 13:55 - 14:00 PST
The Zoom details for the meeting are as follows:
Link: https://up-edu.zoom.us/j/95002648258
Meeting ID: 950 0264 8258
Passcode: KUUPB0801

Closing