Related papers: The Quartic Residues Latin Square

Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$ is at least 2 and $U$ is the union formed by taking certain arithmetic progressions…

Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.

Let $D$ be an irreducible plane curve. In this article, we first introduce a notion of a quadratic residue curve mod $D$, and study quadratic residue concis $C$ mod an irreducible quartic curve $Q$. As an application, we study a dihedral…

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with…

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo…

We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv…

The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and \varpi=2.62205... is the real period of the elliptic…

Let $\varepsilon>0$ be a fixed small constant, ${\mathbb F}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb F}_p$ that involve intervals and arbitrary sets. Representative…

For Latin squares the units (rows and columns) have fixed sum. The same holds for rows, columns, and blocks in Sudokus. Summing the elements of a unit yields a linear equation, and the set of all such equations forms a system of linear…

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. In the present paper, we construct the Hilbert genus…

We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial.…

The multplicative order of an integer g modulo a prime p, with p coprime to g, is defined to be the smallest positive integer k such that g^k is congruent to 1 modulo p. For fixed integers g and d the distribution of this order over residue…

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4…

We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares…

We show that every sum of squares in the three-variable Laurent series field $\mathbb{R}((x,y,z))$ is a sum of 4 squares, as was conjectured in a paper of Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that every…

We study quark and lepton mass matrices with the common modulus $\tau$ in the $A_4$ modular symmetry. The viable quark mass matrices are composed of modular forms of weights $2$, $4$ and $6$. It is remarked that the modulus $\tau$ is close…

We will see that every finite projective plane of order k > 1 gives rise to a complete set of (k-1) MPLS (= mutually projective latin squares) of order k and by reversing the process we can construct a finite projective plane of order k…