Can You Solve This Prison Inmate's Viral Math Riddle?

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Last year, an inmate serving 25 years for murder made headlines by publishing an academic paper in the journal Research in Number Theory

As a Popular Mechanics report explains, Christopher Havens dropped out of high school, but started teaching himself mathematics during solitary confinement shortly after starting his 2011 sentence.

Specifically, Havens became enthralled with the field of number theory, which delves into the study of integers and their functions.

In prison, Havens reportedly used to read the "Problems" section of Math Horizons, an undergraduate-level mathematics publication. Now, Math Horizon is printing one of Havens' own math problems. The problem the prison inmate submitted reads as follows:

'What is the smallest positive integer y such that 1729y2+1 is a perfect square?'

The problem posed by Havens references a famous story related to Indian mathematician and number theorist Srinivasa Ramanujan, who was born on December 22, 1887.

In a conversation between Ramanujan and University of Cambridge number theorist G.H. Hardy, the latter told Ramanujan that he had taken a taxi with the number 1729.

According to the story, Hardy remarked that the number was particularly dull, to which Ramanujan reportedly replied: "No, it’s is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Ramanujan noticed in almost an instant that the number 1729 can be written as both 13+123 and 93+103.

Pell's equation and the chakravala method

Havens' problem is an example of Pell's equation, also known as the Pell–Fermat equation. It is of the form x2−Ny2=1 where N is a positive nonsquare integer.

As Popular Mechanics points out, one method for solving Pell's equation was found about 500 years before the equation was wrongly attributed to English mathematician John Pell — Leonhard Euler attributed another contemporary's solution to the equations to Pell, but the name stuck and was never corrected.

Indian mathematician Bhāskara II, who lived in the 12th century, devised an algorithm for solving Pell's equation — it is known as the chakravala method.  

The idea at the root of the chakravala method is to start with a guess of a solution and adapt it incrementally in order to eventually find the correct solution.

Finding the solution

Another algorithm that can be used to solve Pell's equation has problem solvers find the continued fraction representation of the square root of the coefficient (N) in the equation. So in Christopher Havens's problem, 1792. Similar to the chakravala method, continued fractions are approximations. 

As Evelyn Lamb writes in her article for Popular Mechanics:

"As the height of the tower of numerators and denominators grows, the continued fraction approximation gets closer to the irrational number being approximated. The insight of the continued fraction approach to solving Pell’s equation is that when x and y are large, a difference of 1 is relatively small. In other words, numbers that satisfy x2−Ny2=1 are close to being numbers that satisfy x2=Ny2, or (x/y)2=N. Hence looking for a rational number x/y whose square is close to 1729 will help you find numbers x and y that satisfy x2−1729y2=1."

In order to get from the continued fraction for √1729 to the solution for Pell's equation, you must use the rational approximation (called a convergent) derived at each step, written as a fraction x/y. You then have to see if the convergent satisfies the equation x2−1729y2=1.

This is slow methodical work that is well suited to someone experiencing the boredom of solitary confinement. Want to check if you have the correct solution, or simply take a quicker root to the answer? Simply type in 1792 in this Pell's equation calculator. 

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