Diophantine equations in semiprimes

TL;DR

This paper establishes conditions under which certain homogeneous polynomial equations have infinitely many solutions where each coordinate is a semiprime, extending previous results on solutions with bounded prime factors.

Contribution

It provides new sufficient conditions similar to Birch's criteria ensuring infinitely many solutions with semiprime coordinates for homogeneous equations.

Findings

Conditions for solutions with semiprime coordinates are established.

Extension of previous results from bounded prime factors to semiprimes.

Infinitely many solutions exist under the new criteria.

Abstract

A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, \ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to those of the seminal work of B. J. Birch, for which the equation $F (x_1, \ldots, x_n) = 0$ has infinitely many integer solutions with semiprime coordinates. Previously it was known, by a result of \'A. Magyar and T. Titichetrakun, that under the same hypotheses there exist infinitely many integer solutions to the equation with coordinates that have at most $384 n^{3/2} d (d+1)$ prime factors.