FARGO, N.D. – Facing stiff resistance, President Bush began searching state-by-state for support for his plan to overhaul Social Security and conceded yesterday that not all lawmakers believe the program has a serious problem.

“The math doesn’t work,” Mr. Bush insisted, saying Social Security would pay out more money than it brought in beginning in 2018. “And in 2042, it’s bust,” he said. That’s the year in which the system would be able to cover only about 73% of benefits owed unless it is changed, according to Social Security trustees.

Mr. Bush spoke at the Bison Sports Arena at North Dakota State University, the first stop on a two-day, five-state trip to try to build support for diverting some Social Security revenues into private investment accounts for younger workers. The initiative would reduce guaranteed retirement benefits but create the possibility of bigger checks from stock market investments.

Meanwhile, Mayor Bloomberg spoke out against the president’s Social Security plan yesterday while at a groundbreaking ceremony at Brooklyn.

“I’ve never thought this was a great idea,” Mr. Bloomberg said. “I think the Social Security system has served this country well and it does have its problems and has never been a system that saved for the future. But I don’t think privatization is the solution to that problem.”

Mr. Bloomberg told the New York Times that he did not believe he was taking sides against Mr. Bush or his party, but was instead restating one of his own long-held positions on the issue.

In Fargo, Mr. Bush continued his plea for both parties to cooperate. “We’re not going to play politics with the issue,” Mr. Bush said. “We’re going to say, ‘If you’ve got a good idea, come forth with your idea.’ Because now is the time to put partisanship aside and focus on saving Social Security for young workers.”

But politics played a part in his trip to North Dakota and Montana yesterday and Nebraska, Arkansas, and Florida today. Each state is represented in the Senate by at least one Democrat who GOP strategists believe might back Mr. Bush’s Social Security makeover plan.

Mr. Bush hoped to reach out to Senator Conrad of North Dakota, who is a top-ranking Democrat on the subcommittee overseeing Social Security. “You take what the president said in his address last night, he’s talking about more tax cuts, more spending, more borrowing for private accounts,” Mr. Conrad said in a phone interview. “It doesn’t add up.”

Mr. Conrad said at least five Republican senators have approached him in the last 10 days looking to form a bipartisan group that can offer an alternative to the president’s politically risky proposal.

Like Mr. Bloomberg, many Republicans are nervous about tinkering with the 70-year-old government retirement program, and Mr. Bush needs to convince them as well as skeptical Democrats.

As evidence that Mr. Bush’s plan has prominent hurdles to clear in Congress, opponents of the idea are shadowing the president on his travels, with ads and rallies to drum up public support against the accounts.

In outlining his plan Wednesday night in his State of the Union address, Mr. Bush said private accounts would provide younger workers with retirement money above the check they’d get from Social Security, but he didn’t mention that the check would be smaller.

“The plan that’s been outlined, they know it can’t pass,” Mr. Conrad said.