## Diophantine equations in semiprimes. (arXiv:1709.03605v1 [math.NT])

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 that of the seminal work of B. J. Birch, for which the equation \$F (x_1, \ldots, x_n) = 0\$ has infinitely many solutions whose coordinates are all semiprimes. Previously it was known due to \'A. Magyar and T. Titichetrakun that under the same hypotheses there exist infinite number of integer solutions to the equation whose coordinates have at most \$384 n^{3/2} d (d+1)\$ prime factors. Our main result reduces this bound on the number of prime factors from \$384 n^{3/2} d (d+1)\$ to \$2\$.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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 that of the seminal work of B. J. Birch, for which the equation \$F (x_1, \ldots, x_n) = 0\$ has infinitely many solutions whose coordinates are all semiprimes. Previously it was known due to \'A. Magyar and T. Titichetrakun that under the same hypotheses there exist infinite number of integer solutions to the equation whose coordinates have at most \$384 n^{3/2} d (d+1)\$ prime factors. Our main result reduces this bound on the number of prime factors from \$384 n^{3/2} d (d+1)\$ to \$2\$.