He falsely claimed the Democrats “made the Russian story up as a hoax, as a ruse, as an excuse for losing an election.”

The United States government sounded the alarms long before Mr. Trump was elected. Publicly, in October 2016, the Obama administration formally accused the Russian government of interfering in the election. Privately, at a Group of 20 meeting in September 2016 in China, President Barack Obama told President Vladimir V. Putin of Russia to “cut it out.” And John O. Brennan, the former C.I.A. director, said he also warned his Russian counterpart in a phone call that August.

He claimed to have “saved coal,” contrary to trends reported by the government.

The number of coal mining jobs fell to about 48,600 in September 2016 from almost 90,000 in January 2012. Under Mr. Trump, employment in the sector made modest gains until this September — when job numbers reached about 51,700 — but it has since declined.

Coal production, too, increased in the first quarter of 2017, but declined in the second quarter. The Energy Information Administration, in its latest assessment of the coal sector, estimated “lower exports and no growth in coal consumption” in 2018.

He overstated his influence on the special Senate election in Alabama.

Mr. Trump claimed that he brought Luther Strange, the Republican appointed to fill the Senate seat vacated by Jeff Sessions, now the attorney general, “up by 20 points” during the Republican primary race for the special Senate election in Alabama. But that assertion is not supported by the data.

Before Mr. Trump endorsed Mr. Strange for the August vote, Mr. Strange placed secondbehind Roy S. Moore by two and eight points in two polls, and led the field by two points in another. He was not in “fifth” among the candidates in the Republican primary race, as Mr. Trump said. During the subsequent runoff election in September, Mr. Strange trailed Mr. Moore by an average of 11 points in polls and eventually lost by 9.2 percentage points.

“We can concretely say that Donald Trump’s endorsement, and active campaigning for Sen. Strange, had absolutely no impact on the ballot,” Firehouse Strategies, a Republican polling firm, wrote after the runoff.

He gave a premature estimate of the cost of the wars in the Middle East.

The $7 trillion cost, “as of about a month ago,” that Mr. Trump cited appears to refer to an assessment from Brown University that tallies war appropriations, increases in the Pentagon’s war budget, veterans’ care, increases in spending at the Department of Homeland Security and interest payments. Researchers at Brown estimated in September that war spending had reached $4.8 trillion and could total $7.9 trillion by 2053.

He falsely claimed to have “essentially gutted and ended Obamacare.”

As part of the tax law, Republicans repealed the Affordable Care Act’s so-called individual mandate that required that most people have health insurance, but that does not amount to a full repeal of the current health care law.

The mandate is, indeed, a core component of the Affordable Care Act. But other vital parts of the current law — the expansion of Medicaid eligibility, rules stipulating insurance policies cover essential health benefits and new taxes to pay for the cost of subsidized coverage — remain intact.

With no evidence, he accused other countries of sending their “worst people” through the diversity visa lottery.

Mr. Trump correctly noted that the man accused of killing eight people in a terrorist attack in October in New York entered the United States from Uzbekistan through the diversity immigration visa program. But he was wrong to draw the sweeping conclusion that other countries use the program to expel the “bad, worse” members of their societies.

Foreign government do not select entrants, but rather millions of individuals enter the lottery of their own volition. A computer chooses winners at random and, before receiving a visa, those selected must undergo a screening process that bars criminals and the indigent.

Immigrants who have been admitted through the program have higher rates of employment and of finding work in professional or managerial occupations than most other immigrants.

He exaggerated the trade deficit with other countries.

The figures that Mr. Trump listed account only for the United States’ trade in goods and do not include trade in services. In 2016, the United States’ trade balance amounted to a $309 billion deficit with China (not $350 billion); a $63 billion deficit with Mexico (not $71 billion); and a $7.7 billion trade surplus with Canada (not a deficit of $17 billion).

In Mr. Trump’s telling, Prime Minister Justin Trudeau of Canada disputed the president’s estimate of a $17 billion deficit when they discussed the issue behind closed doors, but he excluded oil and lumber to claim a balance. The trade balance data does, in fact, include oil and lumber.

He exaggerated the number of followers he has on social media.

Mr. Trump, who began the year by falsely claiming the largest inaugural crowd ever, ended it by playing up the extent of his support on social media. As of Friday, Mr. Trump has 45.4 million followers on Twitter, 24.2 millionon Facebook and 8.1 million on Instagram, for a total of 77.7 million, less than half of the 158 million he claimed. (The White House’s accounts on those same platforms would bring him up to 106 million.)