On the exceptional set of a system of linear equations with prime numbers
https://doi.org/10.22405/2226-8383-2023-24-2-15-37
Abstract
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.
About the Authors
Ismail Allakov
Termez State University
Uzbekistan
Uzbekistan
doctor of physical and mathematical sciences, professor
Bakhrom Kholturayevich Abrayev
Termez State University
Uzbekistan
Uzbekistan
basic doctoral student
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Review
For citations:
Allakov I., Abrayev B.Kh. On the exceptional set of a system of linear equations with prime numbers. Chebyshevskii Sbornik. 2023;24(2):15-37. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-2-15-37
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