On the exceptional set of a system of linear equations with prime numbers
Abstract
Let πβ be a sufficiently large real number, π1, π2-integers with 1 β©½ π1, π2 β©½ π, πππ ,(π = 1, 2; π = 1, 4)β positive integers, π1,. . ., π4βprime numbers.Let π΅ = max {3 |πππ|} , (π = 1, 2; π = 1, 4), Β―π = (π1, π2), πΎ = 9β2π΅3β βΒ―πββ,πΈ2,4(π) ={οΈππββ1 β€ ππ β€ π, ππ ΜΈ= ππ1π1 + Β· Β· Β· + ππ4π4, π = 1, 2}οΈ.The paper studies the solvability of a system of linear equations ππ = ππ1π1+Β· Β· Β·+ππ4π4, π = 1, 2,in primes π1, . . . , π4 and for the first time a power estimate for the exceptional set πΈ2,4(π) and a lower estimate for π (Β―π)β the number of solutions of the system under consideration in prime numbers, are obtained, namely, that if π is sufficiently large and πΏ(0 < πΏ < 1) is sufficiently small real numbers, then: there exists a sufficiently large number π΄, such that for π > π΅π΄ estimate is fair πΈ2,4(π) < π2βπΏ; and for π (Β― π) given Β―π= (π1, π2), 1 β©½ π1, π2 β©½ π fair estimate π (Β― π) β©Ύ πΎ2βπΏ(lnπΎ)β4, for all Β―π= (π1, π2) except for at most π2βπΏ pairs of them.