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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!

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    $\begingroup$ Do natural numbers include zero? $\endgroup$ – Kenny Lau Jul 31 '16 at 12:43
  • $\begingroup$ There are infinitely many primes $\endgroup$ – Kenny Lau Jul 31 '16 at 12:43
  • $\begingroup$ @KennyLau we spontaneously conclude. $\endgroup$ – Zack Ni Jul 31 '16 at 12:45
  • $\begingroup$ 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 $\ \ $ $\endgroup$ – Bill Dubuque Jul 31 '16 at 23:41
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Hint:

Prime is infinite of form $6k+1$ where $k$ is an integer and prime cannot be written as two product.

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  • $\begingroup$ ...except with 1. $\endgroup$ – Kenny Lau Jul 31 '16 at 12:45
  • $\begingroup$ @kennyLau my typing is so slow so when I go to university, I will train my self lololol $\endgroup$ – Zack Ni Jul 31 '16 at 12:46
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    $\begingroup$ 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. $\endgroup$ – lulu Jul 31 '16 at 13:12
  • $\begingroup$ @lulu that's why my word "Hint" there. It only remind the OP of how to carry out the solution rather than give a full solution and give the OP some clue about how to prove infinitely many primes of the form 6A+1. $\endgroup$ – Zack Ni Jul 31 '16 at 13:22
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    $\begingroup$ @ZackNi I think that "hint" is far too vague. Please give more details about the proof that you have in mind. $\endgroup$ – Bill Dubuque Jul 31 '16 at 15:02

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