I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$ n=6xy\pm x\pm y $$ for $x,y\in\Bbb Z^+$.

[Note: The $\pm$ signs above are independent; therefore, the above equation is actually 4 separate conditions which can be expanded as $$ n=6xy+x+y $$ $$ n=6xy-x-y $$ $$ n=6xy+x-y $$ $$ n=6xy-x+y $$ (for $x,y\in \Bbb Z^+$) or compressed into $$ n=6|x||y|-(x+y) $$ (for $x,y\in \Bbb Z^*$). If $0$ is an allowed value for $x$ or $y$, then there are two trivial solutions for any integer with $x=-n$ or $y=-n$ and the other equal to $0$; therefore, these solutions are ignored by using the set of all non-zero integers, $\Bbb Z^*$.]

To state the problem formally, I want to show that there does not exist an integer ($N$) such that all integers greater than $N$ can be written in the form $n=6|x||y|-(x+y)$. Namely: $$ \forall (x,y,N\in\Bbb Z)\exists (n>N)[(6|x||y|-(x+y)= n)\rightarrow ((x=0\land y=-n)\lor(x=-n\land y=0))]. $$

Besides a proof (or disproof) of this hypothesis, I would also be interested in resources for studying this problem or other advice on how to proceed. Any suggestions for reading material concerning this sort of Diophantine equation would be greatly appreciated. I understand that proof strategies for these sorts of statements often take advantage of the properties of elliptic functions; would that be applicable here?


Edit: For the interested, here is the OEIS page for the integers that satisfy these requirements. http://oeis.org/A067611

  • $\begingroup$ do you have a list of numbers not representable up to, say, 1000? $\endgroup$ – Will Jagy Nov 4 '13 at 1:55
  • $\begingroup$ Here is the OEIS page. oeis.org/A067611 $\endgroup$ – Geoffrey Nov 4 '13 at 1:59
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    $\begingroup$ You are asking to show that the sequence of $n$ such that $6n-1$ and $6n+1$ are both prime. That is, you are asking for a proof that there are infinitely many twin primes. $\endgroup$ – Will Jagy Nov 4 '13 at 2:32
  • $\begingroup$ I am actually aware of this. I was hoping that it wouldn't come up right away because I figured it would turn people off. I'm really just interested in getting a better understanding of the sorts of techniques I might apply to a problem of this sort (namely, the problem of showing that no solutions exist above a certain threshold, or the problem of determining the membership of a number in a solution set when that number is very large). $\endgroup$ – Geoffrey Nov 4 '13 at 6:32

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