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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:&nbsp;</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 &quot;Fermat's Last Theorem, a genetic introduction to algebraic number theory,&quot; confirmed that the discovery was unknown and called it &quot;quite remarkable&quot;. (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 &quot;an expert on Fermat's Last Theorem&quot;, 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.&nbsp; 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 &quot;notes on Fermat's Last Theorem,&quot; 1st edition).</font></FONT><font face="Georgia">&nbsp; 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 &quot;Beal's conjecture,&quot; &quot;Beal's problem,&quot; or the &quot;Beal problem.&quot; </P></FONT> <p>&nbsp;</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> <p> <p>&nbsp;</td> </tr> </table> </center> </div> </BODY></HTML> <!-- FILE ARCHIVED ON 10:27:40 Oct 2, 2016 AND RETRIEVED FROM THE INTERNET ARCHIVE ON 0:03:29 Dec 27, 2016. 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