How can I Find all solutions of the diophantine equation? :
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Given this answer that Bill gave you just one hour ago, this would be unfair to give you another similar one for this question as well.
Hint: (Inspired by a similar hint from Bill Dubuque)
Rewrite the equation as $$(2 x - 1) (2 y - 3) = 3$$ and equate $2x - 1$ with the factors of $3$.