Pergola : Can I use 4 - 6x6 post, with 2x10 beams sandwitching 6x6 post for 17ftx17ft
Assuming the pergola is a “lattice” type roof and not a “solid” roof, (because you mentioned 2x2 slats,) the total load on any one side would be about 1.5 psf for 2x6’s 24” o.c. plus 2.0 psf for 2x2’s 6” o.c. For a total load of about 3.5 psf. 

So, the total load on the 2x10’s would be about 3.5 psf x 17’/2 = about 30 plf. 

Depending on the species and grade of 2x10’s, one 2x10 spanning 17’ will support about 60 plf. So, two 2x10’s spanning 17’ will support about 120 plf, which indicates your support beams are grossly oversized. 

Connection of the 2x10’s to the 6x6’s can be with 3 - 16d, which will support about 100 lbs. each nail. 

Of major concern is how the 6x6’s will be supported and stay plumb.