5

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_5

Source: first round of 1950

Prove that for every positive integer

k

there exists a sequence of

k

consecutive positive integers none of which can be represented as the sum of two squares.

number theory proposednumber theory

Miklos Schweitzer 1950_5

Source: second part of 1950

1\le a_1<a_2<\cdots<a_m\le N

be a sequence of integers such that the least common multiple of any two of its elements is not greater than

N

m\le 2\left[\sqrt{N}\right]

\left[\sqrt{N}\right]

denotes the greatest integer

\le \sqrt{N}

least common multiplenumber theory proposednumber theory