arXiv subjects

Kok Seng Chua

Publications and source records attributed to Kok Seng Chua.

4 recordsLinked to original sources

arXiv2026

Chebyshev polynomials and a refinement of the local residue/non-residue structure at a prime

The basic power function $t_n(x)=x^n$ is in some sense a classical limit for large $x$, of the monictised Chebyshev polynomial of the first kind $T_n(x)/2^{n-1}$. A theorem of Ritt says they are the only two families of polynomials $p_n(x)$ over $\mathbb{C}$ which satisfies the commutativity relation $p_n(p_m(x))=p_m(p_n(x))$. The commutativity $t_n(t_m(x))=t_m(t_n(x))$ is the reason why the RSA scheme allow also digital signature but the Diffie-Hellman key exchange protocol depends only on the commutativity. The DH scheme and many results in elementary local (at a fixed prime) multiplicative number theory is about properties of the power function $t_n(x)$ and they have natural analogue extension to $T_n(x)$. Recently we discovered a Chebyshev version of Euler's primality criterion , which however depends on two quadratic characters $ε_p(a)=\left ( \frac{a^2-1}{p} \right)$ and $δ_p(a)=\left( \frac{2(a+1)}{p} \right)$. This gives rise to a local partition of $(\mathbb{Z}/p\mathbb{Z}) \setminus \{ \pm 1 \}$ into 4 disjoint sets $A_{εδ}$. This can be thought of as a real refinement of the residue/non-residue as it arise from viewing $T_n(x)$ is the "real" part of the $n$th power of the unit $ω_x=x+\sqrt{x^2-1}$, namely $ω_x^n=T_n(x)+U_{n-1}(x)\sqrt{x^2-1}$. There are obvious analogue of Chebyshev version of pseudoprimes, Wieferich primes, Lucas-Lehmer, AKS, Diffie-Hellman, cyclotomic expansions and probably others.

math.GM↗

arXiv2024

Inverting the wedge map and Gauss composition

Let $1 \le k \le n,$ and let $v_1,\ldots,v_k$ be integral vectors in $\mathbb{Z}^n$. We consider the wedge map $α_{n,k} : (\mathbb{Z}^n)^k /SL_k(\mathbb{Z}) \rightarrow \wedge^k(\mathbb{Z}^n)$, $(v_1,\ldots,v_k) \rightarrow v_1 \wedge \cdots \wedge v_k $. In his Disquisitiones, Gauss proved that $α_{n,2}$ is injective when restricted to a primitive system of vectors when defining his composition law for binary quadratic forms. He also gave an algorithm for inverting $α_{3,2}$ in a different context on the representation of integers by ternary quadratic forms. We give here an explicit algorithm for inverting $α_{n,2}$, and observe via Bhargava's composition law for $\mathbb{Z}^2 \otimes \mathbb{Z}^2 \otimes \mathbb{Z}^2 $ cube that inverting $α_{4,2}$ is the main algorithmic step in Gauss's composition law for binary quadratic forms. This places Gauss's composition as a special case of the geometric problem of inverting a wedge map which may be of independent interests. We also show that a given symmetric positive definite matrix $A$ induces a natural metric on the integral Grassmannian $G_{n,k}(\mathbb{Z})$ so that the map $X \rightarrow X^TAX$ becomes norm preserving.

math.NT↗

arXiv2021

A p-adic identity for Wieferich primes

Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ with $gcd(a,b)=1$, $p \nmid ab$, and $p|(a^n \pm b^n)$, we prove the identity $$ν_p(a^n \pm b^n)-ν_p(n)=ν_p(a^{p-1}-b^{p-1}).$$ An unintended interesting immediate consequence is the following variant of Wieferich's criterion for FLT : Let $x^n+y^n=z^n$ with $n$ prime and $x,y,z$ pairwise relatively prime. Then every odd prime $p|y$ satisfies $ν_p(z^{p-1}-x^{p-1}) \ge n-1$ and every odd prime $p|x$ satisfies $ν_p(z^{p-1}-y^{p-1}) \ge n-1$, and every odd prime $p|z$ satisfies $ν_p(x^{p-1}-y^{p-1}) \ge n-1$, ie. every odd prime dividing $xyz$ is a Wieferich prime of order at least $n-1$ to some base pair. In the "first case" where $n \nmid xyz$, the lower bound for the Wieferich order can be improved to $n$. This gives us very strong intuition why there should not be any solution even for moderately large $n$.

math.NT↗

arXiv2021

Chebyshev polynomials and higher order Lucas Lehmer algorithm

We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x \rightarrow x^2-2$ is extended by Chebyshev polynomial $T_n(x)$ with an implied iteration algorithm because of the compositional identity $T_n(T_m(x))=T_{nm}(x)$. This results from a Chebyshev polynomial primality test based essentially on the Lucas pair $(ω_a,\overlineω_a)$, $ω_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. It seems interesting that the arithmetic are all coded in the Chebyshev polynomials $T_n(x)$.

math.NT↗