Diophantine equations in semiprimes

Abstract

A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1,\dots ,x_n)$ 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 $F(x_1,\dots ,x_n)=0$ has infinitely many integer solutions with semiprime coordinates. Previously it was known, by a result of Á. Magyar and T. Titichetrakun, that under the same hypotheses there exist infinitely many integer solutions to the equation with coordinates that have at most $384n^{3/2}d(d+1)$ prime factors.

21 pages

References in corpus (1)

• Diophantine equations in the primes