Small prime solutions of linear ternary equations

Small solutions of linear Diophantine equations

Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <...

A Numerical Bound for Baker's Constant - Some Explicit Estimates for Small Prime Solutions of Linear Equations

proved that there is an absolute constant V > 0 such that the linear equation a 1 p 1 + a 2 p 2 + a 3 p 3 = b has prime solutions p j 's if b (max j a j) V and a j > 0. Apart from the numerical value of V , the bound is sharp. In this manuscript, we obtain a numerical bound for V. We also obtain a numerical bound for the small prime solutions of the above equation if the a j 's are not all of t...

Weakly prime ternary subsemimodules of ternary semimodules

In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...

Solutions of Linear Equations

For simplicity, we follow the rules: x denotes a set, i, j, k, l, m, n denote natural numbers, K denotes a field, N denotes a without zero finite subset of N, a, b denote elements of K, A, B, B1, B2, X, X1, X2 denote matrices over K, A′ denotes a matrix over K of dimension m × n, B′ denotes a matrix over K of dimension m × k, and M denotes a square matrix over K of dimension n. We now state a n...

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations