# What natural numbers are not equal to the sum of the sum and the product of two natural numbers

What natural numbers $n$ do not satisfy the equation

$$n = (x+y)+xy$$

where $x$ and $y$ are both natural numbers?

• Is $0$ a natural number? – Micah Nov 25 '14 at 6:39
• The text we're using has natural numbers greater than 0, so no. Thanks for the hint! – Jesse Nov 25 '14 at 7:24

Hint: $xy +(x+y)= (x+1)(y+1)-1$
If 0 is a natural number, then every $n$ can satisfied $n=x+y+xy$, since you can let $x=n,$ and $y=0$.
If 0 is not a natural number, notice that $x+y+xy=(x+1)(y+1)-1$, then the request $n$ can be, for example 3,5,7,9, $\cdots$, $2k+1$, $\cdots$; however, it contains other numbers, for example, 8,14,$\cdots$