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Scrooge McDuck in Mathmagic land

A sympathetic story for you this Saturday.

Andy has a problem. He can’t solve it on his own – he needs your help. This problem vexed Andy so much that he spent four years trying to solve it on his own, to no avail. It really is a very difficult problem. Finally in 1997, out of what must have been sheer desperation, Andy reached out to his fellow man: maybe some kindly type out there could find a solution to his problem, which he would gladly reward with a small consideration.

Can you help a soul in need?

His name is D. Andrew “Andy” Beal, billionaire. He owns a mansion and a yacht1. He’s a businessman from Dallas Texas, he’s worth 8.5 billion dollars, and he has previously spent that money on playing poker, almost launching rockets into space and avoiding prosecution for filing phony tax losses.

The news story is, “Andrew Beal, billionaire and mostly terrible person, will pay money to whoever solves this problem, which he thinks he thought of first but actually didn’t, all despite the complete lack of evidence, or even in spite of the abundance of evidence to the contrary, that mathematicians are motivated or even can be motivated by monetary reward.”

In 1997 an article written by Texas mathematician R. Daniel Mauldin was published in the Notices of the AMS, announcing both the problem and a prize of \$5,000 for solving it, offered by Andrew Beal. The problem is this:

Let $A$, $B$, $C$, $x$, $y$ and $z$ be positive integers, with $x,y,z \gt 2$. If \[ A^x + B^y = C^z,\] then $A$, $B$ and $C$ have a common prime factor.

It’s a fairly old problem and usually called the Tijdeman-Zagier conjecture, but Beal likes to call it “The Beal Conjecture”. Nobody has yet claimed the prize, and Beal has increased the prize fund several times over the years. If you’ve been sitting on a solution, maybe a million dollars can tempt you to reveal it? Last week, it was announced that Beal has increased the prize to \$1,000,000.

If you don’t have a solution but the prospect of a million dollars has bought your attention, first of all what’s wrong with you, and secondly, an overview paper by Noam Elkies on $ABC$-type problems is a good place to start your investigations.

An unattributed website on the conjecture and prize, bealconjecture.com, is registered to “Beal Aerospace Technologies, Inc.”, a company which was going to do private satellite launches. bealconjecture.com contains a digression complaining about Granville et al’s comments re the attribution of the conjecture, and bealaerospace.com contains two letters whinging about NASA and that hard-working schmoes like Dennis Tito had to pay Russia to have a go on the ISS.

It isn’t unheard of for prizes to be offered to solve a mathematical conjecture – in the late 19th century the King of Sweden offered a prize for a solution to the $n$-body problem; there are the Clay Millennium prizes, offered by another American businessman and famously turned down by Perelman; and Erdős very often handed out small amounts of money for solutions to problems. What those all have in common is that the bounty offered didn’t increase, and what I object to with Beal is this idea he apparently has that he just needs to find the market price of a solution and it will appear.

Take it away, Pink Floyd!

More information

The Beal Prize information page at the American Mathematical Society.

Beal’s own page about the conjecture.

A question titled “Status of Beal, Granville, Tijdeman-Zagier Conjecture” at MathOverflow.

A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem by R. Daniel Mauldin in Notices of the AMS.

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