Given Mr. Trump’s refusal to release his returns, it remains unclear whether he has managed to pay no taxes. Mr. Trump suggested after the debate that viewers would be mistaken to draw that conclusion from his comments onstage.

“Of course I pay federal taxes,” he told reporters.

Though it is not required by law, every other presidential nominee from a major party since 1976 has released tax returns, or at least a summary of them, and the public has pored over the paperwork for clues about how wealthy and charitable the candidates are. In recent years, with rising concerns about income inequality and the emergence of ultrarich candidates, those returns have also been scrutinized for whatever breaks officeseekers might be using to avoid paying the higher rates expected of the top 1 percent.

When Mitt Romney was the Republican nominee in 2012, his return from the previous year revealed that his effective tax rate had been 14 percent, strikingly low for someone of his wealth. (At the time, the top marginal tax rate was 35 percent.)

Mr. Romney and his wife, Ann, even gave up $1.75 million in charitable deductions that they were entitled to, in order to make their tax return consistent with Mr. Romney’s earlier claim that he had paid at least 13 percent of his income in taxes.

Mr. Romney, who had a long career in finance, benefited from the lower tax rate on so-called carried interest, which can make up a substantial portion of a private equity manager’s income. Democrats in Congress have tried unsuccessfully to tax carried interest as ordinary income. Though Mr. Trump’s campaign has declined to answer questions about his taxes and tax rates, there are several ways tax law could be helping him, experts said.

Tax write-offs are generous at the start of any business project, because some ventures do not start making money for years. But when it comes to treatment by the tax code, “real estate developers have it better than anyone else,” said Steven M. Rosenthal, a tax lawyer and senior fellow at the Urban-Brookings Tax Policy Center.

There was a point when even ruinous projects like an unfinished, unleased office tower could end up producing a profit, thanks to ample tax write-offs.

When President Ronald Reagan and Congress agreed to a tax overhaul in 1986, those loopholes were mostly closed for outside investors — including lawyers, dentists and small-business owners — who put money in such developments for the sole reason of generating losses on paper that would enable them to shelter their other income from taxes. But active real estate investors and developers were allowed to keep that tax break, Mr. Rosenthal said, and the power of the real estate lobby has muted any thought of revoking it.

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Mitt Romney, the Republican nominee in 2012, and his wife, Ann, were revealed to have had a low effective tax rate. Credit Stephen Crowley/The New York Times

Limited liability companies, of which Mr. Trump has scores, and partnerships can also deduct the cost of the interest they pay on loans, a benefit for sectors like real estate that depend on debt financing. (House Speaker Paul D. Ryan, Republican of Wisconsin, included a proposal to end interest payment deductions in his tax blueprint.)

Though little is known about Mr. Trump’s recent income taxes, limited information about his taxes in some years has surfaced in court and regulatory records.

In a 1981 report assessing Mr. Trump’s fitness for a casino license, New Jersey’s casino commission reported that he had paid more than $71,000 in federal income taxes on about $218,000 of taxable income earned from 1975 to 1977.

But in 1978 and 1979, the report said, Mr. Trump paid no federal income taxes. By taking advantage of deductions available to real estate developers and claiming losses from partnerships, Mr. Trump reported a “negative income” of $406,379 in 1978 and $3.4 million in 1979 — thus avoiding any tax liability for those two years, a time when he claimed to be worth hundreds of millions of dollars.

Tax court records indicate that Mr. Trump also avoided paying any federal income taxes in 1984. In 1991 and 1993, at a time when Mr. Trump’s Atlantic City casinos were in deep financial trouble, casino commission reports show that he claimed losses that would have allowed him to avoid paying income taxes in those years, too. Mr. Trump may have been able to use those losses to reduce or eliminate his federal tax bill for years to come.

With the race in its final phase, there is substantial interest among voters in Mr. Trump’s making his taxes public, according to recent polls. A CBS News/New York Times Poll this month showed that 59 percent of respondents said it was necessary for him to release his tax returns. In a survey by Fox News last month, 60 percent said they believed Mr. Trump was hiding something in his returns.

Mr. Trump says he is not releasing his tax returns because they are under a routine audit by the Internal Revenue Service, but officials from that agency have said he is under no obligation to keep his forms secret. In an interview this month with The Pittsburgh Tribune-Review, Mr. Trump’s oldest son, Donald Jr., offered a different explanation, saying that a release of the tax returns, which he said covered 12,000 pages, would “distract” from his father’s message.

Mrs. Clinton and her husband, former President Bill Clinton, have released their 2015 return and have noted, almost proudly, the taxes they paid. The forms showed that they paid $3.6 million in federal taxes on adjusted gross income of $10.6 million, for a 35 percent tax rate. Even so, the Clintons, like many high-income Americans, have made use of trusts that reduce their estate tax burden and would allow their heirs to inherit more of their wealth.

During the debate, Mrs. Clinton seized on Mr. Trump’s comments, saying that if he had not paid taxes, that meant “zero for troops, zero for vets, zero for schools or health.”

Grover Norquist, the president of Americans for Tax Reform and a prominent Trump supporter, said in an interview that Mrs. Clinton’s line of attack was unfair.

“It is a dirty trick of her to pull that,” Mr. Norquist said. “She knows damn well that people in that industry, housing and construction, lose millions of dollars or gain millions of dollars in a year,” based on a number of factors, sometimes outside the control of the developer.

Lost in the debate is that Mr. Trump did not fail to pay taxes that he owed, Mr. Norquist said. “He didn’t owe any taxes.”

In hindsight, Mr. Norquist said, Mr. Trump should have simply said, “I’ve paid every dollar in taxes I owed.”

But Bob McIntyre, the director of Citizens for Tax Justice, which favors tax policies in which the wealthy pay a larger share, called Mr. Trump’s seeming pride in paying little or nothing in taxes an “anti-American” view.

He said he worried that Mr. Trump’s supporters would take his words as inspiration to pay no taxes themselves, legally or otherwise, because they “hate the government and think they are going to waste your money.”

“Think where we would be if everyone did that,” Mr. McIntyre said.

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