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“Donald hates his birthdays,” Ivana Trump once said of her ex-husband.

He seemed to hate this one, on Friday, especially. He looked genuinely upset on Sunday when his supporters sang an early “Happy Birthday” to him at a rally in Las Vegas. “You know,” he told them, “there’s a certain point at which you don’t want to hear ‘Happy Birthday.’ You just want to pretend the day doesn’t exist.”

No doubt. Now 78, he is trying to convince the country to give him another four years in office, at the end of which he would be the oldest president in American history — a mantle held by the current, 81-year-old occupant of the Oval Office.

On the campaign trail, both men try to downplay their age while making the other appear as creaky and cranky as possible.

The Trump team spent the week blasting out selectively edited videos of Mr. Biden toddling around Europe, looking like Corrado Soprano lost in Newark. “Biden was seen staring off into the distance and wandering around like a brain-dead zombie,” said Karoline Leavitt, a Trump campaign press secretary.

The Biden campaign hit back with an email on Mr. Trump’s birthday reminding voters that he fell asleep at his own trial, a case that resulted in felony convictions. “He is unhinged, unable to focus and diminishing before our very eyes,” James Singer, a Biden campaign spokesman, said of the former president.

Mr. Trump used to have fun on his birthdays. “Donald was the child who would throw the cake at the birthday parties,” his younger brother Robert once told the journalist Marie Brenner. After he became a mogul, Mr. Trump would throw swinging parties at his casinos in Atlantic City, like the one at Trump’s Castle in 1988, for his 42nd birthday, when he played videos of Liza Minnelli and Billy Crystal wishing him a happy birthday. (President Ronald Reagan sent a telegram.)