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<p align="center"><img src="images/beallogo.gif" alt="The Beal Conjecture" border="0" WIDTH="288" HEIGHT="43"></p>
--><FONT
face=Georgia color=#800000 size=6>
<B>
<P align=center>The Beal Conjecture</B></FONT><font face="Georgia"> </font> </P><STRONG>
<FONT face=Georgia
color=#008080>
<P>Background</P></FONT></STRONG><FONT face=Georgia>
<P>Mathematicians have long been intrigued by Pierre Fermat's famous assertion
that A<SUP>x</SUP> + B<SUP>x</SUP> = C<SUP>x</SUP> is impossible (as stipulated)
and the remark written in the margin of his book that he had a demonstration or
"proof". This became known as Fermat's Last Theorem (FLT) despite the lack of a
proof. Andrew Wiles proved the relationship in 1994, though everyone agrees that
Fermat's proof could not possibly have been the proof discovered by Wiles.
Number theorists remain divided when speculating over whether or not Fermat
actually had a proof, or whether he was mistaken. This mystery remains
unanswered though the prevailing wisdom is that Fermat was mistaken. This
conclusion is based on the fact that thousands of mathematicians have
cumulatively spent many millions of hours over the past 350 years searching
unsuccessfully for such a proof.</P>
<P>It is easy to see that if A<SUP>x</SUP> + B<SUP>x</SUP> = C<SUP>x</SUP> then
either A, B, and C are co-prime or, if not co-prime that any common factor could
be divided out of each term until the equation existed with co-prime bases.
(Co-prime is synonymous with pairwise relatively prime and means that in a given
set of numbers, no two of the numbers share a common factor.)</P>
<P>You could then restate FLT by saying that A<SUP>x</SUP> + B<SUP>x</SUP> =
C<SUP>x</SUP> is impossible with co-prime bases. (Yes, it is also impossible
without co-prime bases, but non co-prime bases can only exist as a consequence
of co-prime bases.)</P></FONT><STRONG>
<FONT face=Georgia color=#008080>
<P>Beyond Fermat's Last Theorem</P></FONT></STRONG>
<P><font face="Georgia">No one suspected that<b> </b></font><STRONG>
<FONT face=Georgia
color=#008080>A<SUP>x</SUP> + B<SUP>y</SUP> = C<SUP>z</SUP></FONT></STRONG><FONT face=Georgia>
</FONT><FONT face=Arial,geneva>
<font face="Georgia">(note unique exponents) might also be impossible with
co-prime bases until a remarkable discovery in 1993 by a Dallas, Texas number
theory enthusiast by the name of D. Andrew Andy Beal. (Refer to
<a href="/web/20161002102740/http://www.andrewbeal.com/">www.andrewbeal.com</a> for more information
about Andy Beal.) Andy Beal was working on FLT when he
began to look at similar equations with independent exponents. He constructed
several algorithms to generate solution sets but the very nature of the
algorithms he was able to construct required a common factor in the bases. He
began to suspect that co-prime bases might be impossible and set out to test his
hypothesis by computer. Andy Beal and a colleague programmed 15 computers and after
thousands of cumulative hours of operation had checked all variable values
through 99. Many solutions were found: all had a common factor in the bases.
While certainly not conclusive, Andy Beal now had sufficient reason to share his
discovery with the world.</font></P>
<p><b><font face="Georgia">BEAL'S CONJECTURE: </font></b></FONT><FONT face=Georgia>
</FONT><FONT face=Arial,geneva> <font face="Georgia">If A<sup>x</sup>
+ B<sup>y</sup> = C<sup>z</sup>, where A, B, C, x, y and z are positive integers
and x, y and z are all greater than 2, then A, B and C must have a common prime
factor.</p>
<p>In the fall of 1994, Andy Beal wrote letters about his work to
approximately 50 scholarly mathematics periodicals and number theorists. Among
the replies were two considered responses from respected number theorists.<br>
<br>
Dr. Harold Edwards from the department of mathematics at New York University and
author of "Fermat's Last Theorem, a genetic introduction to algebraic number
theory," confirmed that the discovery was unknown and called it "quite
remarkable". (Edward s response,
<a target="_blank" href="Beal_Conjecture_Response_Harold_M_Edwards.pdf">PDF</a>) <br>
<br>
Dr. Earl Taft from the department of mathematics at Rutgers University relayed
Andy Beal's discovery to Jarell Tunnell who was "an expert on Fermat's Last
Theorem", according to Taft's response, and also confirmed that the discovery
and conjecture were unknown. (First Response, <a target="_blank" href="Beal_Conjecture_Response_Earl_Taft_and_Jerrell_Tunnell_2_pages.pdf">PDF</a>;
Second Response,
<a target="_blank" href="Beal_Conjecture_Response_Earl_Taft_Second_Response.pdf">
PDF</a>) </p>
<p>There is no known evidence of prior knowledge of Beal's conjecture and all
references to it begin after Andy Beal's 1993 discovery and subsequent
dissemination of it. The related ABC conjecture hypothesizes that only a finite
number of solutions could exist.</p>
<p><strong><font color="#008080" face="Georgia">Interesting Side Note</font></strong></p>
<p><font face="Georgia">While Beal's conjecture was widely received with
enthusiasm by the mathematics community at large, it seems that there are always
people with other motivations.</font></p>
<p><font face="Georgia">Three mathematicians by the names of Andrew Granville, Alf
Vanderpoorten, and Andrew Bremmer set out to discredit and criticize Andy Beal and
the attribution of the conjecture, including writing scathing criticisms
claiming that the conjecture was previously known and evidenced in prior works
dating back to Brun's work of 1914. The untruthfulness and motivations of their
criticisms were made obvious by the subsequent 1996 publication of
Vanderpoorten's book wherein he claims to propose the conjecture as recent
original thought by none other than & . himself (see page 194 of Vanderpoorten's
"notes on Fermat's Last Theorem," 1st edition).</font></FONT><font face="Georgia">
Equally interesting, the Beal conjecture has more recently (after 2010, almost
20 years after Beal's dissemination of the conjecture) also been incorrectly
referred to as the Tijdeman-Zagier conjecture, although no known evidence exists as to when or
where Tijdeman or Zagier ever conjectured or disseminated any such conjecture.
Apparently Vanderpoorten himself also would contest the Tijdeman-Zagier
attribution as evidenced by his 1996 publication referenced above. The fact
remains that while many people may claim they made the discovery, no evidence
has ever been produced predating Beal's discovery and widespread dissemination
of the conjecture in 1993.</p>
</font><STRONG>
<FONT face=Georgia
color=#008080>
<P>Encouraging Others</P></FONT></STRONG><FONT face=Georgia>
<P>By offering a cash prize for the proof or disproof of this important number
theory relationship, Andy Beal hopes to inspire young minds to think about the
equation, think about winning the offered prize, and in the process become more
interested in the wonderful study of mathematics. Information regarding the
$1,000,000 cash prize that is held in trust by the American Mathematics Society
can be obtained at the University of North Texas web site: </FONT><A
href="/web/20161002102740/http://www.math.unt.edu/~mauldin/beal.html"><FONT
face=Georgia>http://www.math.unt.edu/~mauldin/beal.html</FONT></A><FONT
face=Georgia>.</P>
<P>Incidentally, Andy Beal believes that the world has yet to adequately respond to
Fermat's challenge to the English in 1657 regarding Pell's equation (see Edwards
book referenced above - book section 1.9). Andy Beal believes that Fermat may well
have had a method of solution that was "not inferior to the more celebrated
questions of geometry in respect of beauty, difficulty, or method of proof".
Fermat claimed his method involved infinite descent and no known methods of
solution use a descent. Furthermore, the continued fraction and cyclic methods
of solution known today hardly qualify as beautiful or particularly difficult.
For someone seeking to learn about or advance number theory, that's a great and
fun place to start.</P>
<P>The Beal Conjecture is sometimes referred to as "Beal's conjecture," "Beal's
problem," or the "Beal problem." </P></FONT>
<p> </td>
</tr>
<tr>
<td width="100%"><b><font color="#800000">NEWS ABOUT THE BEAL CONJECTURE</font></b><ul>
<font FACE="Georgia" COLOR="#5a5b5e"><i>
<li></i></font>
<a HREF="/web/20161002102740/http://abcnews.go.com/blogs/headlines/2013/06/billionaire-offers-1-million-to-solve-math-problem/">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>ABCNews: Billionaire Offers
$1 Million to Solve Math Problem</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.businessinsider.com/beale-conjecture-1-million-dollar-prize-2013-6">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Business Insider: If You
Can Solve This Math Problem, A Texas Banker Will Give You $1M</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.dallasnews.com/news/education/headlines/20130604-dallas-banker-andrew-beal-ups-ante-for-solving-tricky-math-problem-to-1-million.ece">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Dallas Morning News: Dallas
banker Andy Beal ups ante for solving tricky math problem to $1M</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.huffingtonpost.com/2013/06/05/andrew-beal-math-problem_n_3389601.html">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Huffington Post: Andy Beal
Offers $1M To Solve Problem That's Stumped Since The 1980s</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://newsfeed.time.com/2013/06/11/solve-this-math-problem-win-a-million-bucks/">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Time: Solve This Math
Problem, Win a Million Bucks</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.upi.com/Science_News/2013/06/08/Math-group-increases-reward-for-proving-the-Beal-Conjecture/UPI-24731370724763/">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>UPI: Math group increases
reward for proving the Beal Conjecture</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.usnews.com/news/offbeat/articles/2013/06/04/texas-banker-puts-up-1m-for-tricky-math-solution">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>USNews: Texas banker puts
up $1M for tricky math solution</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.isciencetimes.com/articles/5338/20130605/1-million-math-problem-banker-d-andrew.htm">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>iScienceTimes: $1 Million
Math Problem: Banker Andy Beal Offers Award To Crack Conjecture
Unsolved For 30 Years</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://blogs.nature.com/news/2013/06/mathematics-prize-ups-the-ante-to-1-million.html">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Nature: Mathematics prize
ups the ante to $1M</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.sciencerecorder.com/news/texas-banker-offers-1-million-for-solution-to-insane-math-problem/">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>Science Recorder: Texas
banker offers $1 million for solution to insane math problem</i></u></font></a>
<font FACE="Georgia" COLOR="#5a5b5e"><i></li>
<li></i></font>
<a HREF="/web/20161002102740/http://www.andrewbeal.com/andybeal/media/medialibrary/Documents/Beal-Conjecture-Prize-Increased-130603.pdf">
<font FACE="Georgia" COLOR="#5a5b5e"><u><i>AMS Press Release re: Beal
Conjecture Prize</i></u></font></a></p>
</ul>
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