Quadratic polynomials at prime arguments

详细信息    查看全文

• 作者：Jie Wu ; Ping Xi
• 关键词：Quadratic polynomial ; Almost prime ; Greatest prime factor ; Primes in arithmetic progressions ; Sieve method
• 刊名：Mathematische Zeitschrift
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：285
• 期：1-2
• 页码：631-646
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1823
• 卷排序：285

文摘

For a fixed quadratic irreducible polynomial f with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes p such that f(p) has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by \(P^+(n)\) the greatest prime factor of n, it is also proved that \(P^+(f(p))>p^{0.847}\) infinitely often.