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“You can’t tell whether there’s something problematic, but it’s pretty striking that this one was written down so much,” said David Skeel, a professor at the University of Pennsylvania Law School who specializes in bankruptcy law and reviewed the case at the request of The New York Times.

The refusal by Trump, the Republican presidential nominee, to release his personal income tax returns has become a growing issue in the campaign. He has also boasted of his success in lowering his tax burden as a businessman, declaring last year in an interview on Fox News that only “a stupid person, a really stupid person, is paying a lot of taxes.”

By that measure, the deal with New Jersey looks remarkably shrewd. The casinos did far better, for example, than those that benefited from a program Christie introduced in 2014 in which the state agreed to consider reducing penalties for delinquent taxpayers but only if they caught up on all overdue taxes and interest.

Public records do not create a clear picture of how the agreement was reached. A spokeswoman for Trump said she would be in touch regarding questions sent to her.

But she did not reply further or respond to subsequent messages.

Brian Murray, a spokesman for Christie, said the governor had not been aware of the tax dispute and, therefore, could not comment on the terms of the settlement.

The Times discovered the agreement during a review of the thousands of documents filed in the bankruptcies of Trump’s casinos. The taxes went unpaid from 2002 through 2006, during which time Trump was leading the company as chairman and, until 2005, as its chief executive. He reaped millions of dollars in fees and bonuses from the company, even as it underperformed competitors, lost money every year and saw its stock collapse.