MathDB

Consider the set

A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.

[*]Prove that every integer

x \geq 2

can be written as the product of one or more elements of

A

, which are not necessarily different.
[*]For every integer

x \geq 2

let

f(x)

denote the minimum integer such that

x

can be written as the product of

f(x)

elements of

A

, which are not necessarily different. Prove that there exist infinitely many pairs

(x,y)

of integers with

x\geq 2

y \geq 2

, and

f(xy)<f(x)+f(y).

(Pairs

(x_1,y_1)

and

(x_2,y_2)

are different if

x_1 \neq x_2

y_1 \neq y_2

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