On a Diophantine Inequality with s Primes

Let 2 < c < δ . In this study, for prime numbers p 1 , … , p s and a sufficiently large real number N , we prove the Diophantine inequality p 1 c + ⋯ + p s c − N < N − 9 / 10 c δ − c , where δ = 294 − 210 s / 123 − 97 s and s ≥ 5 . When s = 5 , this result improves a previous result.

On the Multiplicity of a Proportionally Modular Numerical Semigroup

A proportionally modular numerical semigroup is the set S a , b , c of nonnegative integer solutions to a Diophantine inequality of the form a x mod b ≤ c x , where a , b , and c are positive integers. A formula for the multiplicity of S a , b , c , that is, m S a , b , c = k b / a for some positive integer k , is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a , b , c such that k ≠ 1 when the number a − c is fixed.

A Diophantine inequality with four prime variables

Let [Formula: see text] be a sufficiently large real number. In this paper, it is proved that, for [Formula: see text], the following Diophantine inequality [Formula: see text] is solvable in prime variables [Formula: see text], which improves the result of Mu [On a Diophantine inequality over primes, Adv. Math. (China) 44(4) (2015) 621–637].

A ternary Diophantine inequality involving primes

Let [Formula: see text]. In this paper, it is proved that for every sufficiently large real number [Formula: see text], the Diophantine inequality [Formula: see text] is solvable in primes [Formula: see text]. This result constitutes an improvement upon that of Baker and Weingartner.