# Does the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ?

How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers differing by $4$ )

• What is a "powerful number"? May 14, 2015 at 15:55
• @GFauxPas : en.wikipedia.org/wiki/Powerful_number
– user228168
May 14, 2015 at 15:56
• Try looking up Pell's equation. May 14, 2015 at 16:07

If $(x,y)$ is a solution, then so is $(5x+4y, 6x+5y)$. Starting with $(1,1)$, you get a sequence of solutions. What is $x$ mod $3$ for these?

• the $x$'s are not coming to be $0$ mod $3$ , if I assume $3|5x+4y$ , then $5(5x+4y)+4(6x+5y) \equiv x+y$ (mod $3$ ) .... what should I do then ?
– user228168
May 14, 2015 at 16:16
• Iterating the transformation mod $3$ I get $(1,1) \to (0,2) \to (2,1) \to (2,2) \to (0,1) \to \ldots$. May 14, 2015 at 19:36

Put another way, $$27 \cdot 3^2 - 2 \cdot 11^2 = 1.$$ Given one $(x,y)$ pair with $$27 x^2 - 2 y^2 = k,$$ we get the next pair, of infinitely many, from $$(485x + 132 y, 1782x+485 y).$$

With your $k=1$ you need start with the single seed pair $(3,11)$ to get all $(x,y)$ pairs with $x,y>0.$

For composite $k,$ you may need two or more seeds, but a small finite number in any case.

• where did those $485 , 132 , 1782$ come from ?
– user228168
May 15, 2015 at 4:51