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Functional Family: Mock theta mystery solved Erica Klarreich A pair of mathematicians has solved a problem that had tantalized number-theory researchers for more than 8 decades. It is the so-called final problem of the legendary Indian mathematical genius Srinivasa Ramanujan. In the years before his death in 1920, Ramanujan studied theta functions, which are numerical relationships that show special symmetries. On his deathbed, Ramanujan wrote a letter to his British collaborator G. H. Hardy, in which he listed 17 complicated formulas for new functions. He called them mock theta functions because they had some properties similar to those of theta...

At 28, Peter Scholze is uncovering deep connections between number theory and geometry. In 2010, a startling rumor filtered through the number theory community and reached Jared Weinstein. Apparently, some graduate student at the University of Bonn in Germany had written a paper that redid “Harris-Taylor” — a 288-page book dedicated to a single impenetrable proof in number theory — in only 37 pages. The 22-year-old student, Peter Scholze, had found a way to sidestep one of the most complicated parts of the proof, which deals with a sweeping connection between number theory and geometry. “It was just so stunning...

A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave. o mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan...

The Math Genius Who Gave Us Sudoku By AMY ELLIS NUTT   Leonhard Euler's math discoveries extend to astronomy, ballistics, optics, music theory, fluid dynamics, mapmaking, shipbuilding — and Sudoku.     In the spring of 1727, two weeks after the body of Isaac Newton was laid to rest in London's Westminster Abbey, an obscure 19-year-old mathematician by the name of Leonhard Euler left his home in Basel, Switzerland, to take up an academic position in St. Petersburg, Russia.When he died there in 1783 at the age of 76, Euler (pronounced "oiler'') had become the most prolific scientific writer in...

In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. Will the subatomic world help reveal the elusive nature of the primes?In 1972, the physicist Freeman Dyson wrote an article called "Missed Opportunities." In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell's equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough—only announced by Einstein some 40 years later. It is striking that Dyson should have written about...

<p>Working with a Turkish colleague, a San Jose State University math professor has solved one of the most important problems in prime number theory -- a solution that took him 20 years.</p> <p>Mathematicians described the advance -- announced at a conference in Germany -- as the most important breakthrough in the field in decades. Like many mathematical developments it has no immediate practical application but may open the door to a wealth of further advances.</p>