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The GOP would then need to pick up six seats, net, to win the Senate.

As it stands, political handicappers say the GOP should capture open seats in West Virginia, South Dakota and Montana — three red states that GOP presidential nominee Mitt Romney won in the 2012 election.

They also say control of the Senate likely hinges on whether Republicans can oust three of the four most vulnerable Democrats — a list that includes Mr. Pryor as well as Sens. Mary L. Landrieu in Louisiana, Mark Begich in Alaska and Kay R. Hagan in North Carolina.

The math, though, could change.

Democrats hope that underdog Alison Lunderland Grimes can knock off Sen. Minority Leader Mitch McConnell in Kentucky and that Michelle Nunn, the daughter of former Democratic Sen. Sam Nunn, can emerge victorious in race to fill an open seat in Georgia.

Whatever the case, Mr. Cotton’s announcement is good news for Republicans. A Harper Polling/Conservative Intel poll this week found that Mr. Cotton held a 43 percent to 41 percent edge over Mr. Pryor.

Nathan L. Gonzales, deputy editor of The Rothenberg Political Report, said Mr. Cotton is “a credible candidate” and gives “Republicans a good chance of winning state.”

Kyle Kondik of the University of Virginia’s Center for Politics said that while Mr. Pryor has some tough votes to defend, he enjoys all the advantages of incumbency and has tried to put some distance between himself and national Democrats.

The latest campaign finance reports, filed June 30, show Mr. Pryor has more than $3.9 million cash on hand, while Mr. Cotton has more than $1 million.