Yitang Zhang

New Delhi, May 24: A Chinese-American mathematician who once worked as an accountant with a sandwich outlet has taken a big step towards proving a conjecture about prime numbers that has challenged mathematical minds for over a century and a half.

Yitang Zhang at the University of New Hampshire has proved what mathematicians call a weak version of the twin-prime conjecture, whose strong version states that there are an infinite number of primes only two numbers apart — such as 3 and 5, 17 and 19, or 641 and 643.

Zhang has shown that the number of consecutive primes whose difference is less than 70 million is infinite. While the gap of 70 million appears huge compared with the difference of 2, mathematicians say his proof is a major advance.

“This is an exciting development. This is the first real evidence in support of the (twin-prime) conjecture,” said Gadadhar Mishra, a senior mathematician at the Indian Institute of Science, Bangalore, who was not associated with this work.

Zhang, who had worked on the problem for nearly four years, said the idea of how to tackle it had emerged last year while he was visiting a friend, an orchestra conductor in Colorado, hoping to enjoy some classical music.

“I was trying to relax completely — I had not taken any of my books or papers — but I couldn’t stop thinking about mathematics,” Zhang told The Telegraph over the phone. His proof has been accepted for publication in the journal Annals of Mathematics.

“This is a big feat. What is particularly fascinating is that Zhang has done this cleverly using existing (mathematical) techniques rather than by inventing something new,” said Ramachandran Balasubramanian, director of the Institute of Mathematical Sciences, Chennai.

“The twin-prime conjecture holds that twin primes persist forever no matter how far down we go the number line,” Balasubramanian said. “But this has to be rigorously proven and Zhang has taken a major step towards that.”

The Greek mathematician Euclid had proved the infinitude of prime numbers around 300 BC, and French mathematician Alphonse de Polignac had formalised the twin-prime conjecture in 1849. Since then, many mathematicians had tried to find ways of developing techniques to prove the conjecture.

Zhang was born in Shanghai and went to college in Beijing before moving to the US for a PhD in mathematics. While waiting for a suitable opening in an academic institution, Zhang said, he had to work as an accountant with a sandwich outlet in Lexington, Kentucky.

He joined the University of New Hampshire in 1999 where he has been teaching calculus, including differential equations, and complex analysis to undergraduate and graduate students.

“There are several unsolved problems involving the prime numbers,” Zhang said, indicating he might wish to pursue some other problem rather than look to complete the search for a proof of the twin-prime conjecture.

Balasubramanian said there had been a sense among number theory researchers that the mathematical techniques needed to solve the twin-prime conjecture had not been developed yet.

According to a Wikipedia entry, the largest pair of twin primes found so far have 200,700 digits each.