The world’s favorite number is seven, at least if the result of a poll conducted by Alex Bellos is to be believed. Some people like it because it is prime, some because they have a lot of sevens in their birthdates. But I went to a talk by 2014 Fields Medalist Manjul Bhargava that gave me another reason to love this number: exponential Diophantine equations.
Diophantine equations, named for the mathematician Diophantus of Alexandria, are equations relating several unknown quantities in which we are only concerned with solutions that are integers. For example, if we look for Pythagorean triples, integers that satisfy the equation a2+b2=c2, we're looking at a Diophantine equation. Logically enough, exponential Diophantine equations are Diophantine equations in which one of the unknowns is an exponent.
In 1913, Indian mathematician Srinivasa Ramanujan conjectured that the exponential Diophantine equation 2n-7=x2 has a solution in which n and x are both integers only when n=3,4,5,7, and 15. In 1948, Norwegian mathematician Trygve Nagell proved it, although in response to his countryman Wilhelm Ljunggren, not Ramanujan.
2n-7=x2 seems like a rather arbitrary equation to study, and it turns out to be special. The equation 2n-D=x2 has at most two solutions for any (nonzero) D other than 7. There are two puzzling things about this result. First, why is seven special? Second, did Ramanujan know that seven was special? If not, why did he pick that equation out of all the possibilities? I haven't been able to find an answer to that.
When there's one special case for a number, it seems like the number is usually 0 or 1. Maybe 2 every once in a while. But it seems like 7 rarely gets to be special. It usually has to settle for the people's choice award instead.
Bhargava's told us about much more than this number theory nugget in his talk, a video of which is available on the Heidelberg Laureate Forum website. It is the same talk he gave at the International Congress of Mathematicians where he received the Fields Medal. It is a beautiful, accessible presentation, well worth watching.
This blog post originates from the official blog of the 2nd Heidelberg Laureate Forum (HLF) which took place in Heidelberg, Germany, September 21 – 26, 2013. 24 Abel, Fields, and Turing Laureates gathered to meet a select group of 200 young researchers.