# Who Wants to be a Millionaire?

It is rare to find reportings of math in the mainstream media that explain an actual algebra equation. It is rarer still to find such reportings that involve a math problem attached with serious prize money. The other day I found just such a piece in time.com – a math problem with an award of \$1 million (US dollars). I would like to discuss this math problem and its significance.

It turns out this is a number theory problem that is intimately connected to the famous math problem called Fermat’s Last Theorem. So that is where we start.

Fermat’s Last Theorem
There are no positive integers $A$, $B$, $C$ and $x$ that satisfy the following equation

$\displaystyle A^x+B^x=C^x$

where $x$ is greater than 2.

A bit of perspective. The requirement that $x>2$ is necessary. If $x$ = 2, then the equation $\displaystyle A^2+B^2=C^2$ would have many solutions. For examples, $\displaystyle 3^2+4^2=5^2$ and $\displaystyle 5^2+12^2=13^2$ (note that this is the familiar Pythagorean Theorem).

Fermat’s Last Theorem was stated as a true mathematical fact by Pierre de Fermat in 1637 (at least he implied that he had a proof). He scribbled in the margin of his own copy of the Greek text Arithmetica by Diophantus commenting that a proof of this statement was too large to fit in the margin.

No one knows whether Fermat really had a proof of his conjecture. But we do know that no one could prove Fermat's last Theorem until 1995. For several hundred years (until 1995), Fermat's Last Theorem was regarded as one of the most difficult mathematical problems.

In 1993 Andrew Wiles, a British mathematician, had formulated a proof of Fermat's Last Theorem after working for 6 years in secret. But a flaw was discovered in one part of the proof. In 1995, after working with a former student for over a year, Wiles finally had patched the hole that was in the original proof. Thus Fermat’s Last Theorem had ceased to be a conjecture and became a true mathematical statement.

Apparently the successful resolution of Fermat’s Last Theorem was not the end of the story. Soon after, people began to hunt for more research opportunities by generalizing Fermat’s Last Theorem. Someone proposed a new problem, which is now called Beal’s Conjecture (after Andrew Beal, the person who proposed it). Consider the following statement.

Beal’s Conjecture
There are no positive integers $A$, $B$, $C$, $x$, $y$ and $z$ that satisfy the following equation

$\displaystyle A^x+B^y=C^z$

where

$x$, $y$ and $z$ are greater than 2, and

$A$, $B$ and $C$ share no common prime factor.

Note the similarity between this new statement and Fermat’s Last Theorem. In fact, Fermat’s Last Theorem is a specific case of Beal’s Conjecture. If the exponents are made the same (and greater than 2), we come back to Fermat’s Last Theorem and we know that the equation $\displaystyle A^x+B^x=C^x$ has no integer solutions (as proven by Andrew Wiles).

However, when the exponents are not equaled, the equation $\displaystyle A^x+B^y=C^z$ does have integer solutions. To illustrate, note that $\displaystyle 3^3+6^3=3^5$. Here, $A$, $B$ and $C$ share the common prime factor 3. Another example: $\displaystyle 7^6+7^7=98^3$. Now $A$, $B$ and $C$ share the common prime factor 7. Beal’s Conjecture is essentially saying that if the exponents $x$, $y$ and $z$ are greater than 2, the equation $\displaystyle A^x+B^y=C^z$ has integer solutions only when $A$, $B$ and $C$ share a common prime factor (just like in the 2 examples just given).

Andrew Beal, the one who proposed the Beal’s Conjecture in 1993, is a Texas billionaire banker who is also an enthusiastic amateur mathematician. After unsuccessful attempts at resolving his conjecture, he put up prize money to spur interest – \$5,000 in 1997 and \$100,000 in year 2000. He upped the prize money just recently to \$1 million in June 2013.

Just in case there is any reader who still thinks the prize is a joke or a hoax, keep in mind that Andrew Beal is a prominent banker with financial worth in the billions. In fact, the \$1 million prize money is already in safe keeping at the American Mathematical Society (AMS). Here’s is more information about the Beal’s Prize. Mr. Beal would also like to use this prize to draw young people into the wonderful world of mathematics. According to AMS, “one of Andrew Beal’s goals is to inspire young people to think about the equation, think about winning the offered prize, and in the process become more interested in the field of mathematics“.

Beal’s conjecture is no doubt a significant math problem because of its connection with Fermat’s Last Theorem. The serious prize money attached to it will give it exposure and generate some buzz. It is no doubt a difficult problem, clearly not a one-weekend project and clearly not a problem that can be solved by just using elementary math techniques (but applied in a clever way). It took over 350 years to resolve Fermat’s Last Theorem. So it will likely take years before we see any progress on Beal’s Conjecture (the conjecture is only 16 years old).

Math problems like Beal’s Conjecture are not a fool’s errand. Many new mathematical ideas and methods will often be created in the process of resolving monumental problems such as Fermat’s Last Theorem and hopefully in the case of Beal’s Conjecture, thus pushing forward the frontier of mathematics. No doubt this is one of the goals of Mr. Andrew Beal in setting up the Beal’s prize.