MathDB

Let

N

be a positive integer. Let

R

denote the smallest positive number that is the sum of

m

terms

\sum^m_{i=1}{\pm \sqrt{a_i}}

, where each

a_i, i=1,\cdots, m

is an integer not larger than

N

. Prove that

R\le C\cdot N^{-m+\frac{3}{2}}

for some positive real number

C

.
Proposed by Navid
(Clarification: note that the constant is allowed to depend on

m

but should be independent of

N

, i.e. the equation

R(m,N)\le C(m)\cdot N^{-m+\frac{3}{2}}

should hold for all positive integers

N

)

Problem Statement