I have currently given Prof. Burnet 1631 orbs, how many do I need to get landorus? And do I just need to do one search with Diamond/Pearl/Platinum/Heart Gold/ Soul Silver and then I unlock it?
You can find how to get all extensions here. You unlock Eureka Extension γ (Landorus's Extention) by collecting 1600 more Dream Orbs after you catch Thundurus. You don't need to buy anything for the Alpha, Beta, or Gamma Eureka Extentions, you just need to collect the orbs.
Hope I helped!