An Optimal Bound for Sum of Square Roots of Special Type of Integers∗

نویسندگان

  • Jianbo Qian
  • Cao An Wang
چکیده

The sum of square roots of integers problem is to find the minimum nonzero difference between two sums of square roots of integers. Let r(n,k) denote the minimum nonzero positive value: |∑i=1 √ ai −∑i=1 √ bi|, where ai and bi are positive integers not larger than integer n. We prove by an explicit construction that r(n,k) = O(n−2k+ 3 2 ) for fixed k and any n. Our result implies that in order to compare two sums of k square roots of integers with at most d digits per integer, one might need precision of as many as (2k− 2 )d digits. We also prove that this bound is optimal for a wide range of integers, i.e., r(n,k) = Θ(n−2k+ 3 2 ) for fixed k and for those integers in the form of n = (2k−1 2i )2 (n′+2i)+ (2k−1 2i+1 )2 (n′+2i+1), where n′ is any integer satisfied the above form and i is any integer in range [0,2k−1].

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Upper bounds on the solutions to n = p+m^2

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

متن کامل

On the minimum gap between sums of square roots of small integers

Let k and n be positive integers, n > k. Define r(n, k) to be the minimum positive value of | √ a1 + · · ·+ √ ak − √ b1 − · · · − √ bk| where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. Define R(n, k) to be − log r(n, k). It is important to find tight bounds for r(n, k) and R(n, k), in connection to the sum-of-square-roots problem, a famous open problem in com...

متن کامل

Finding the Smallest Gap between Sums of Square Roots

Let k and n be positive integers, n > k. Define r(n, k) to be the minimum positive value of | √ a1 + · · ·+ √ ak − √ b1 − · · · − p bk| where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. It is important to find a tight bound for r(n, k), in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bou...

متن کامل

Neural Network Modelling of Optimal Robot Movement Using Branch and Bound Tree

In this paper a discrete competitive neural network is introduced to calculate the optimal robot arm movements for processing a considered commitment of tasks, using the branch and bound methodology. A special method based on the branch and bound methodology, modified with a travelling path for adapting in the neural network, is introduced. The main neural network of the system consists of diff...

متن کامل

On Comparing Sums of Square Roots of Small Integers

Let k and n be positive integers, n > k. Define r(n, k) to be the minimum positive value of | √ a1 + · · ·+ √ ak − √ b1 − · · · − √ bk| where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. It is an important problem in computational geometry to determine a good upper bound of − log r(n, k). In this paper we prove an upper bound of 2O(n/ logn), which is better tha...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2006