Hi

There are various ways to 'solve' that equation, If we just asked to solve it like

Clear[x, y]; Solve[13 x^2 + 1 == y^2, {x, y}, Integers]

We will get results like

{{x -> ConditionalExpression[-(((649 - 180 Sqrt[13])^
      C[1] - (649 + 180 Sqrt[13])^C[1])/(2 Sqrt[13])), 
    C[1] \[Element] Integers && C[1] >= 0],

which isn't very good, this is telling me that there could be an infinite number of solutions and that we need to add limits for x and y, so we could ask this

Clear[x, y]; Solve[
 13 x^2 + 1 == y^2 && 0 < x < 10^10 && 0 < y < 10^10, {x, 
  y}, Integers]

{{x -> 180, y -> 649}, {x -> 233640, y -> 842401}, {x -> 303264540, 
  y -> 1093435849}}

We could also ask this

Clear[x, y]; FindInstance[13 x^2 + 1 == y^2, {x, y}, Integers, 2]

{{x -> -98475707056574514234147687427309858128133932917550159192926078\
9721440877383880976428933444437557685123880, 
  y -> -35505921118005041063603236141349722598219803651103555766541663\
27589596496957941863373110484725700882173601}, {x -> 
   1187828648027618759642502802027120177356001334838507471515621279589\
8286867954206065456648483958433444781041798143695420315713621488215926\
4598844841133760412740750545423228379788307197484815120271546811447749\
7159652848682155570439974210170748217810526303751922055046373538138102\
0231971292940018133726241543544132192067186548069882096119540954036932\
577807293981659210167396951286006140, 
  y -> 428277709692864682878803963175563587116293964276476908497499183\
3145609033134526228869899336669731155196749772066652590596524997159085\
2519889096595069946728890727914390171744935742586153090418212768714274\
7310625391649304220246973177349306393132073180127067871356808850048354\
1013054379064542178999843193995372857640443443929933173325176356363542\
2922385906493065904876438405625228363849}}

Hope that helps a little