Integer solutions of $2x+3y=n$

Question

What is the smallest $m$ such that for all $n\geq m$, the equation $2x+3y=n$ has solutions with $x,y \in \mathbb{Z}$ and $x,y\geq2$?

My approach. We can write the solutions in terms of the parameter $t$ as: $$x(t)=-n-3t$$ and $$y(t) = n+2t$$

Setting both of those greater than or equal to $2$, we have $-n-3t\geq2$ and $n+2t\geq2$. Solving for $t$, we find $\dfrac{-n-2}{3} \geq t \geq\dfrac{2-n}{2}$. Solving the resulting inequality, we get $$-2n-4\geq6-3n$$ So $n\geq10$. This seems to be right. For $n=10$, take $(2,2)$. But, suppose $n=11$. Then, $2x+3y=11$ clearly has no solution where both $x,y\geq2$. Is my solution incorrect? Can someone point me in the right direction?

EDIT: Thanks for the great answers!

Answer 1

We consider two cases: $n$ is even and $n$ is odd.

If $n$ is even then $2x+3y = n \implies$ $y$ is an even number. The minimum possible value of even $n$ would be then $2(2)+3(2)=10$. For all even $n \ge 10$, we can set $x = n/2-3$ and $y=2$ to satisfy $2x+3y=n$.

If $n$ is odd then $2x+3y=n$ implies that $y$ is odd. Consequently, $y \ge 3$ and the minimum possible value of odd $n$ is $2(2)+3(3) = 13$. For odd $n \ge 13$, we set $y=3$ and $x= (n-9)/2$ to satisfy $2x+3y=n$.

For above, we conclude that for $n \ge 12$, we can find an integer solution to $2x+3y=n$ for $x\ge2$ and $y \ge 2$.

Answer 2

The answer is clearly $10$ more than the smallest $m'$ such that for all $n'\ge m'$ there are nonnegative integers with $2x'+3y'=n'$. The numbers of the form $2x'+3y'$ for $x'$, $y'\ge0$ are $0,2,3,4,5,\ldots$, not including $1$, but including all even $n'\ge0$ and all odd $n'\ge3$. So the least $m'$ is $2$, and the least $m$ in your problem is $12$ (you can't express $11$ under your constraints).

Answer 3

You are almost there.

Solving for $t$, we find $\dfrac{-n-2}{3} \geq t \geq\dfrac{2-n}{2}$. Solving the resulting inequality, we get $$-2n-4\geq6-3n$$ So $n\geq10$.

You want to find $n$ such that there is at least one integer $t$ satisfying $$\dfrac{-n-2}{3} \geq t \geq\dfrac{2-n}{2}\tag1$$

For this, solving $\frac{-n-2}{3}\ge\frac{2-n}{2},$ i.e. $-2n-4\ge 6-3n$ is not enough.

If $\frac{-n-2}{3}-\frac{2-n}{2}\ge 1$, i.e. $n\ge 16$, then $(1)$ has at least one integer $t$.

For $n=11$, there is no $(x,y)$.

$2\cdot 3+3\cdot 2=12$.

$2\cdot 2+3\cdot 3=13$.

$2\cdot 4+3\cdot 2=14$.

$2\cdot 3+3\cdot 3=15$.

So, the answer is $12$.