On the exceptional set of a system of linear equations with prime numbers

Let 𝑋− be a sufficiently large real number, 𝑏1, 𝑏2-integers with 1 ⩽ 𝑏1, 𝑏2 ⩽ 𝑋, 𝑎𝑖𝑗 ,(𝑖 = 1, 2; 𝑗 = 1, 4)− positive integers, 𝑝1,. . ., 𝑝4−prime numbers.Let 𝐵 = max {3 |𝑎𝑖𝑗|} , (𝑖 = 1, 2; 𝑗 = 1, 4), ¯𝑏 = (𝑏1, 𝑏2), 𝐾 = 9√2𝐵3⃒ ⃒¯𝑏⃒⃒,𝐸2,4(𝑋) ={︀𝑏𝑖⃒⃒1 ≤ 𝑏𝑖 ≤ 𝑋, 𝑏𝑖 ̸= 𝑎𝑖1𝑝1 + · · · + 𝑎𝑖4𝑝4, 𝑖 = 1, 2}︀.The paper studies the solvability of a system of linear equations 𝑏𝑖 = 𝑎𝑖1𝑝1+· · ·+𝑎𝑖4𝑝4, 𝑖 = 1, 2,in primes 𝑝1, . . . , 𝑝4 and for the first time a power estimate for the exceptional set 𝐸2,4(𝑋) and a lower estimate for 𝑅(¯𝑏)− the number of solutions of the system under consideration in prime numbers, are obtained, namely, that if 𝑋 is sufficiently large and 𝛿(0 < 𝛿 < 1) is sufficiently small real numbers, then: there exists a sufficiently large number 𝐴, such that for 𝑋 > 𝐵𝐴 estimate is fair 𝐸2,4(𝑋) < 𝑋2−𝛿; and for 𝑅(¯ 𝑏) given ¯𝑏= (𝑏1, 𝑏2), 1 ⩽ 𝑏1, 𝑏2 ⩽ 𝑋 fair estimate 𝑅(¯ 𝑏) ⩾ 𝐾2−𝛿(ln𝐾)−4, for all ¯𝑏= (𝑏1, 𝑏2) except for at most 𝑋2−𝛿 pairs of them.

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