MathDB

Problem 6

Part of 2015 Postal Coaching

Problems(2)

Composition of Number Theoretic Functions

Source: India Postals 2015 Set 3

11/7/2015
Let k∈Nk \in \mathbb{N}, let xkx_k denote the nearest integer to k\sqrt k. Show that for each m∈Nm \in \mathbb {N}, ∑k=1m1xk=f(m)+mf(m)+1\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}, where f(m)f(m) is the integer part of 4m−3−12\frac{\sqrt{4m-3}-1}{2}
number theoryfunction
Numbers that are sums of squares and cubes

Source: India Postals 2015

11/15/2015
Show that there are infinitely many natural numbers which are simultaneously a sum of two squares and a sum of two cubes but which are not a sum of two 6−6-th powers.
number theory