On December 1, the Utah Republican Party filed a federal lawsuit to regain the type of primary system Utah had before the 2014 session of the legislature changed the law. The old system provided that no one could get on a primary ballot unless the candidate had shown substantial support at a party caucus. The new system allows a petition method for candidates to get on a primary ballot even if they had not shown such support at a party meeting. The case is Utah Republican Party v Herbert, 2:14cv-876-BCW. Here is a copy of the Complaint.
I think the number of signatures should be lowered until there are at least 135 candidates if you really want to pick an arbitrary number.