Goro Shimura, a mathematician whose insights provided the foundation for the proof of Fermat’s Last Theorem and led to tools widely used in modern cryptography, died on May 3 at his home in Princeton, N.J. He was 89.

The death was announced by Princeton University, where Dr. Shimura had been a professor from 1964 until his retirement in 1999.

In 1955, Yutaka Taniyama, a colleague and friend of Dr. Shimura’s, posed some questions about mathematical objects called elliptic curves. Dr. Shimura helped refine Dr. Taniyama’s speculations into an assertion now known as the Taniyama-Shimura conjecture.

But no one knew how to prove it.

The conjecture appeared unconnected to Fermat’s Last Theorem, a seemingly simple statement made by the French mathematician Pierre de Fermat in 1637: Equations of the form an + bn = cn do not have solutions when n is an integer greater than 2 and a, b and c are positive integers. (If n is equal to 2, the statement becomes the Pythagorean theorem, which says that the squares of the lengths of two sides of a right-angled triangle equal the square of the length of the hypotenuse; this equation — a2 + b2 = c2 — has many solutions where all of the numbers are integers. For example, 32+ 42= 52.)