Quadratic equations with five prime unknowns

✍ Scribed by Stephen Kwok-Kwong Choi; Angel V Kumchev

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 230 KB
• Volume: 107
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.03.007

✦ Synopsis

Let b 1 ; ?; b 5 be non-zero integers satisfying gcdðb i ; b j ; b k Þ ¼ 1; for 1piojokp5 and jb j jpjb 5 j for 1pjp5 and n an integer satisfying b 1 þ ? þ b 5 n ðmod 24Þ: In this paper we improve earlier work by M.C. Liu and Tsang and by the first author and J.Y. Liu. In particular, we prove that if b j are not all of the same sign, then the quadratic equation

has prime solutions satisfying p j jj ffiffiffiffiffi jnj p þ jb 1 ?b 4 j 2þe : If all b j are positive then the above equation is soluble provided that ncb 5 ðb 1 ?b 4 Þ 4þe :