His death was announced last week by Yale University, where he was a professor of mathematics.

Working with James W. Cogdell, his main collaborator over a quarter-century, starting in the mid-1970s, Dr. Piatetski-Shapiro shaped a proof of what is known as the Converse Theorem, which finds some deep relationships between different fields of mathematics.

The mathematical constructs are "quite mysterious, even to mathematicians," Dr. Cogdell explained in an interview this week. But the theorem has wide applications, including playing a small but important role in the proof of Fermat's Last Theorem that Andrew Wiles, a Princeton mathematician, completed in 1994.

"It's a very powerful tool," said Peter C. Sarnak, a professor of mathematics at Princeton. "He made sure it's in a form everyone can use it."

Born in Moscow, Ilya Piatetski-Shapiro became interested in mathematics when he was 10. In a memoir, he said that when he learned about negative numbers from his father, a chemical engineer, he was struck "by the charm and unusual beauty" of the concept.

After completing his undergraduate degree in 1951 at Moscow University, he wanted to continue there for his graduate studies. Despite the strong recommendation of Alexander O. Gelfond, a prominent mathematician and member of the Communist Party, the application was rejected.

"That was a time of great anti-Semitism," said Dr. Piatetski-Shapiro's son, Gregory.

Through Gelfond's efforts, he was admitted to the Moscow Pedagogical Institute. But in 1952, he received a letter from government officials ordering him to go to Kazakhstan to teach high school. His parents, fearing that he might be sent to Stalin's labor camps if he refused, told him to go, but he decided not to. After about a year, another letter informed him he did not have to report for the teaching assignment.

During this time, he produced a proof that asserts that certain sequences of integers defined through the power function contain an infinite number of prime numbers. "It was rather unexpected," Dr. Sarnak said.

Dr. Piatetski-Shapiro received his doctorate in 1954.

Dr. Piatetski-Shapiro also attended seminars at the Steklov Mathematical Institute in Moscow, where Prof. Igor Shafarevich influenced his interests toward modern number theory and algebraic geometry.

In 1958, he became a professor of mathematics at the Moscow Institute of Applied Mathematics, and in 1965, he gained an additional professorship at Moscow State University.

Except for a short trip to Hungary, Dr. Piatetski-Shapiro was not allowed to leave the Soviet Union. The authorities told him that he would be free to travel if he joined the Communist Party.

He responded that party membership would distract him from his work.

He lost his job at the Institute of Applied Mathematics, and even his access to mathematical libraries when he signed a petition in support of a dissident colleague. In the early 1970s, the Soviet Union let more Jews emigrate to Israel, including Dr. Piatetski-Shapiro's first wife, Inna, whom he had divorced, and his son. In 1974, he also applied for an exit visa, but was denied.

In 1976, Dr. Piatetski-Shapiro finally received an exit visa. That broke up his second marriage, as his wife remained in Moscow.

Starting in 1977, he divided his time as a professor between Tel Aviv University and Yale.

When Dr. Cogdell was a graduate student at Yale, Dr. Piatetski-Shapiro became his thesis adviser. As part of the oral examination, Dr. Piatetski-Shapiro assigned him some assertions to prove.

Dr. Cogdell, now a professor at Ohio State, recalled that he thought he had found the solutions.

"Then I realized there was a mistake," Dr. Cogdell said. "His response was, 'Very good.' Then I said, 'I know how to fix it,' and he said, 'Even better.' "

Dr. Piatetski-Shapiro is survived by his wife, Edith Piatetski-Shapiro, who lives in New Haven and Tel Aviv; his son, Gregory, of Brookline, Mass.; a daughter, Shelly Shapiro of New York City; a stepdaughter, Niki Lipkin of Tel Aviv; and two grandsons.

As the Parkinson's worsened, Dr. Piatetski-Shapiro relied on others to record notes and he had more trouble talking. In 1992, he decided to stop traveling, but a year or two later, he changed his mind. "He missed all of the contact and finally decided he wouldn't feel better if he traveled or not," Dr. Cogdell said.

His third wife, Edith, a mathematician he had met in Israel, and others helped him walk. His speech fell to a whisper and often he could not talk at all.

For a number of years, when Dr. Piatetski-Shapiro was in a good state, he would phone Dr. Cogdell, and it would be a three-way conversation with Edith repeating Dr. Piatetski-Shapiro's words to Dr. Cogdell and then handing the phone to Dr. Piatetski-Shapiro as Dr. Cogdell replied.

One of the highest honors in mathematics is to be invited to lecture at the International Congress of Mathematics, held only once every four years, and Dr. Piatetski-Shapiro was invited to speak four times, in 1962, 1966, 1978 and 2002, when he was in his 70s and still "knocking off well-known, longstanding problems," said Dr. Sarnak, the professor at Princeton.

Dr. Piatetski-Shapiro was able to deliver the lecture in person twice -- in 1966 in Moscow, and in 1978, after he had immigrated to Israel. For the first invitation, the Soviet Union did not allow him to leave the country and a colleague delivered his speech for him; the last time, he was too tired to attend, and the address was delivered by Dr. Cogdell.

By KENNETH CHANG

Published: March 4, 2009 © New York Times