A day after Republican Dan Sullivan sprung to the lead in Alaska's U.S. Senate race, his opponent, incumbent Democratic Sen. Mark Begich, refused to concede, citing tens of thousands of outstanding votes -- particularly those in rural parts of the state.

Numbers released by Alaska elections officials Wednesday morning showed Begich facing daunting odds: He needs to win a substantial majority of as many as 50,000 uncounted absentee and other outstanding ballots to catch up with Sullivan.

But even as Republicans sounded confident, distributing press clippings touting their substantial cushion, the Begich campaign said it was holding out, issuing a statement shortly before 11 a.m. that was far from a concession.

The statement referenced voters in rural areas, which Begich's campaign had targeted with an unprecedented effort to register voters and bring them to the polls early. As of Wednesday, elections officials hadn't counted any of the early votes from several districts with rural populations.

"Alaskans for Begich is anxious for a final count of all of Alaskans' ballots and respects the procedures, process and timetable of the Alaska Division of Elections," the campaign's statement said, referencing the name of Begich's political committee.

A spokesperson for Begich wouldn't answer questions Wednesday. Begich was with his family and unavailable, another spokesperson said.

With 100 percent of precincts reporting early Wednesday, Sullivan led Begich by about 8,000 votes.

State elections officials said there were nearly 24,000 uncounted absentee and early votes, which won't be tallied until next week. There are also 13,804 absentee ballots voters had requested but not yet returned to the state, though it was unclear how many of those would ultimately end up being counted.

Then there are questioned ballots -- typically cast by Alaskans who voted at the wrong polling place. Elections officials won't know how many questioned ballots were cast until Thursday, but there were roughly 13,000 in the last midterm election in 2010.

To beat Sullivan, Begich would have to take 8,000 votes from Sullivan out of all the outstanding ballots -- which appear unlikely to top 50,000 total. Republicans pointed to the party affiliation of absentee voters, which wasn't substantially different from the party makeup of the general electorate.

And an Alaska Dispatch News review of state elections data at the district level showed that if the roughly 22,000 uncounted early and absentee ballots were distributed the same way Tuesday's votes were, Sullivan would actually add another 1,100 votes to his lead.

"The math is clearly not on Begich's side," Sullivan's campaign manager, Ben Sparks, said in a phone interview.

Sparks said Sullivan was unavailable for an interview Wednesday, and was leaving for training Thursday with the U.S. Marine Corps Reserves.

Even one of Begich's key allies agreed with the Sullivan campaign's assessment. Asked Wednesday morning about Begich's chances of a comeback, Jim Lottsfeldt, who runs the pro-Begich Put Alaska First PAC, said: "I don't think that's going to happen."

"It's, like, too big of a gap," Lottsfeldt said, though he added the gap was likely to shrink.

In 2008, when Begich beat Republican Sen. Ted Stevens -- who was more popular than Sullivan in rural Alaska -- Begich trailed by 3,000 votes on election night, and he ended up winning by 4,000 votes, swinging the results about 2.7 percent.

But that year, more than 100,000 votes were counted in the weeks after the election, and this year, there appears to be only about half that total outstanding, with Begich trailing by 3.6 percent.

Lottsfeldt's pessimistic view was not uniform among Begich's allies, however. Tom Begich, a Democrat who has worked on the state's redistricting maps, said he woke up his brother Wednesday morning to give him new vote totals, which showed Sullivan's lead shrinking by 1,000 from his 9,000-vote edge as the last few precincts were counted.

Tom Begich said he was excited to see absentee votes hadn't yet been counted in several districts where state elections officials had opened dozens of new so-called "in-person absentee" voting places -- where people could vote early.

Begich's campaign invested heavily in boosting turnout in those areas, and some of those districts -- though not all of them -- broke heavily Democratic.

"The odds are against him winning, but it means there's not just a theoretical chance, but a realistic one," Tom Begich said. "I've told Mark I think he's going to win by 113 votes. But we'll see whether that adds up."