“Thank you so much for reporting this,” said Pete Voss, a Facebook spokesman. He could not say why Facebook had not spotted the accounts posing as its top executives, including several that appeared to have existed for more than eight years. “It’s not easy,” he said. “We want to get better.”

Facebook requires people to use their authentic name and identity. Yet the company has estimated that perhaps 3 percent of its users — as many as 60 million accounts — are fake. Some of those accounts are disguised as ordinary people, some pretend to be celebrities such as Justin Bieber.

In congressional testimony this month, Mr. Zuckerberg said Facebook was improving its software to automatically detect and remove such accounts. Facebook officials have said the company blocks millions of fake accounts trying to register each day and analysts said the social network has improved its efforts to remove the accounts.

“Fake accounts, over all, are a big issue, because that’s how a lot of the other issues that we see around fake news and foreign election interference are happening as well,” Mr. Zuckerberg told lawmakers, adding that Facebook is hiring more people to work on reviewing content.

But major holes remain. Interviews with a half-dozen recent victims — and online conversations with nine impostor accounts — showed that the Facebook lottery deception is alive and well, preying particularly on older, less educated and low-income people.

The Mark Zuckerberg and Sheryl Sandberg impostor accounts typically use the executives’ pictures as profile photos and list their Facebook titles. Some post manipulated images of people holding oversize checks. The names of Mr. Zuckerberg and Ms. Sandberg are sometimes misspelled, or use parentheses and middle names (Elliot for Mr. Zuckerberg and Kara for Ms. Sandberg) to evade Facebook’s software.