Question related to Hilbert's 10th problem
As pointed by Yuval and cody there are easy solutions to get infinitely many Diophantine equations that are not provable nor refutable (let's say in PA).

However these syntactic solution result in provably equivalent sets, i.e. sets that the theory can prove they are equivalent. You can consider these as padding arguments. Another way is adding a variable which is not used at all. 

You can also play with adding or removing some strings explicitly (finite variations of the set).

If you want to get Diophantine equations that are "essentially" different (e.g. the sets are not Turing equivalent) then that is more challenging and I think knowing that there is an independent Diophantine equation is not enough, you will need the fact that every r.e. set can be encoded as a Diophantine equation (or something similar).

ps: since you only care independence it is more natural to represent these formulas as Diophantine equations not their negations.