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Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of $b$ can be found. Some $a$ values are $0,2,3,7,11,17....$ I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?

Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of $b$ can be found for a given $a$. Some $a$ values are $0,2,3,7,11,17....$ I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?

Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$?

I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?

Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of $b$ can be found. Some $a$ values are $0,2,3,7,11,17....$ I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?

Are there infinitely many pairs of integers $(a,b)$ such that $2(6a+1)$ divides $6b^2-6ab-b-6a^2-2a-3$? That is, if $a$ is positive and $b$ is negative.

I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?

Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$?

I think that the answer is yes but I have no idea how to show this.

If the answer is yes, are there any polynomial parametric solutions?