On Sums of Powers of Almost Equal Primes
نویسندگان
چکیده
We investigate the Waring-Goldbach problem of representing a positive integer n as the sum of s kth powers of almost equal prime numbers. Define sk = 2k(k − 1) when k > 3, and put s2 = 6. In addition, put θ2 = 19 24 , θ3 = 4 5 and θk = 5 6 (k > 4). Suppose that n satisfies the necessary congruence conditions, and put X = (n/s). We show that whenever s > sk and ε > 0, and n is sufficiently large, then n is represented as the sum of s kth powers of prime numbers p with |p−X| 6 Xk. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.
منابع مشابه
On the Representation of Even Integers as Sum of Two Almost Equal Primes
In this paper we generalize the Chudakov van der Corput Estermann Theorem on the exceptional set in the binary Goldbach problem to a result on the same problem with "almost equal" primes. Actually, we prove that the equation Pi +P2 = 2n is satisfied by almost ali 2ra € [N, 2N] when the primes pi and p-2 lie in the interval [n — U, n + U], with U = n^. Furthermore, we explicitly estimate the num...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کاملSums of Primes and Squares of Primes in Short Intervals
Let H2 denote the set of even integers n 6≡ 1 (mod 3). We prove that when H ≥ X, almost all integers n ∈ H2 ∩ (X,X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
متن کاملOn the symmetry of primes in almost all short intervals
– In this paper we study the symmetry of primes in almost all short intervals; by elementary methods (based on the Large Sieve) we give, for h x log x (c > 0, suitable), a non-trivial estimate for the mean-square (over N < x ≤ 2N) of an average of “symmetry sums”; these sums control the symmetry of the von-Mangoldt function in short intervals around x. We explicitly remark that our results are ...
متن کاملDivisibility Properties of a Class of Binomial Sums
We study congruence and divisibility properties of a class of combinatorial sums that involve products of powers of two binomial coefficients, and show that there is a close relationship between these sums and the theorem of Wolstenholme. We also establish congruences involving Bernoulli numbers, and finally we prove that under certain conditions the sums are divisible by all primes in specific...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014