# Lol, integers

**Number Theory**Level 3

Let \(P_k(a)\) denote the greatest integer n that can not be represented by the form \(ax+by=n\) for at least \(1\) ordered pair (x,y) and a specific \(a\) and \(k\) (\(k\) is defined below in "assumptions and details")

For example, the largest integer n that can't be represented by \(P_4(5)=5x+9y\) is 31 since no integer values of x and y satisfy \(31=5x+9y\)

Find \(P_{2014}(2015)-P_{2013}(2015\))

Tip:

\(P_{2014}(2015)=2015x+4029y\)

\(P_{2013}(2015)=2015x+4028y\)

Assumptions and details

\((a,b,x,y,n)\epsilon \Bbb{N}\)

\(b>a\).

\(b-a=k\)

\(\text{gcd}(a,b)=1\)

This is part of the set Trevor's Ten