# Prove that there are infinitely many natural numbers that can't be written as $6xy+x+y$

Problem: Prove that there are infinitely many natural numbers that can't be written as $6xy+x+y$ where $x,y$ are positive integers.

I noticed that if $A=6xy+x+y$ then $6A+1=(6x+1)(6y+1)$, but I can't proceed from here. A hint would be great!

• Do natural numbers include zero? – Kenny Lau Jul 31 '16 at 12:43
• There are infinitely many primes – Kenny Lau Jul 31 '16 at 12:43
• @KennyLau we spontaneously conclude. – Zack Ni Jul 31 '16 at 12:45
• Since $\,x,y\ge 1\,$ the factors $\,6x+1,6y+1\,$ are both greater than $1$. So if $\,A\,$ has that form then $\,6A+1\,$ is composite. So if $\,6A+1\,$ is prime then $\,A\,$ does not have that form. But there are infinitely many primes of form $\,6A+1.\,$ QED $\ \$ – Bill Dubuque Jul 31 '16 at 23:41

Prime is infinite of form $6k+1$ where $k$ is an integer and prime cannot be written as two product.
• Sorry...don't you need to show that there are infinitely many primes of the form $6A+1$? That is true, of course, but considerably deeper than the infinitude of primes. – lulu Jul 31 '16 at 13:12