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DEMONSTRATIO MATHEMATICAVol. XLNo 22007Xianmeng Meng 1 , Mingqiang WangON ADDITIVE PROBLEMS WITH PRIME NUMBERSOF SPECIAL T Y P EAbstract. Let Pk denote any integer with no more than k prime factors, countedaccording to multiplicity. It is proved that for almost all sufficiently large integers n, satisfying n = 0 or 1 (mod3), the equation n = pi +p\ +p\ has a solution in primes pi,pi,pzsuch that PI + 2 =P2 + 2 = P5, P3 + 2 = P5. It is also proved that for every sufficientlylarge integer M = 0 or 2 (mod 3), the equation M = pi +p\ + p | +p\ + p | has a solution inprimes pi, • • • ,p 5 such that pi + 2 = P 6 , P2 + 2 = P 5 , p 3 + 2 = P 5 , PA + 2 = P2, ps + 2 = P?.1. Introduction and main resultsIn 1923, Hardy and Littlewood [5] conjectured that each integer n canbe written asn = pi + m f + rri2,and Linnik [10] proved that this conjecture is true. As an extension of theGoldbach conjecture, one can consider the following Diophantine equationwith prime variables(1.1)n
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2007
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