**BEAL'S CONJECTURE DISPROVED**

**James Constant **

**math@coolissues.com**

**Beal's Conjecture is disproved for the same reasons Fermat's Last Theorem is proved.**

**For simple proofs see ****http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.htm**

**http://www.coolissues.com/mathematics/BealFermatPythagoras/proofs.htm**

__http://www.coolissues.com/mathematics/Exponential/series.htm__

**Beal's conjecture **

A prize is offered for proof or disproof of Beal's conjecture^{1}, stated as follows: If

(1). . . . . . . . . . . . *x,y,z,m,a,b* *positive integers *. . . . . . *m,a,b>2*

then *x,y,z* have a common factor *>1*.

**Proof of Fermat's Last Theorem**

A proof of Fermat's Last Theorem (FLT) is available using the binomial expansion^{2}. In this proof it is shown that *z *cannot be an integer in the equality

(2) . . . . . . . . . . . . *x,y,z,m positive integers* . . . . . .* **m>2*

(3) . . . . . . . . . . . . *x,y,z,m positive integers** *. . . . . .* **m>2*

(4) . . . .* * . . . . . . . . *x,y,z,m positive integers*. . . . . .*m>2*

and then expressing the parenthesis term as a binomial series, with results

1. Since *m*>2, index *1/m *is not an integer and the series cannot terminate becoming the binomial theorem.

2. The series is absolutely convergent for and divergent for When convergent, the parenthesis term series converges to an irrational number and thus *z* in equation (4) is an irrational number for any *x*^{a/m}. When divergent, the parenthesis term series and thus *z* in equation (4) have no values.

3. For *y*^{b}*/x*^{a}*=+1* the series converges absolutely since *m>2 *and thus *1/m>0.* The same argument applies. In this case , an irrational number except when *x=2**^k**k* integer, *ka>=m-1 *and* ka>=b*.

In summary, Beal's conjecture is disproved for the stated conditions *x,y,z,m,a,b* positive integers and *m,a,b>2*. It is no longer a conjecture but an observation that Beal's equality exists only when *x*^{a}*=y*^{b}*,**x=2**^k**k* integer, *ka>=m-1 *and* ka>=b***. See also Beal's Conjecture And The Problem of Multiplying Irrational Number's at **** http://www.coolissues.com/mathematics/Beal Problem/bealprob.htm**

**Mirabilis Disproof of Beal's Conjecture**

**Write the right hand side of equation (1) as**

(5) *x*^{2a/2}* + y*^{2b/2 } *x,y,a,b integers a,b>2*

which, by Pythagora's theorem, is a real number squared, say

(6) *v*^{2}=*x*^{a}* + y*^{b } **v**^{2} >2 or = 2 when x=y=1

in which *v*^{2} is an integer and v is an integer or irrational number. Equality equation (1) is obtained by assuming

(7) *v*^{2}=*z*^{m} m > 2 positive integer

in which *z*^{m} is an integer and z is an integer or irrational number. Therefore, equality equation (1) cannot meet Beal's conditions^{ }

^{(8) }*x,y,z,m,a,b positive integers **m,a,b>2*^{ }

^{because z is an integer or irrational number. QED}

^{The Mirabilis proof is a partial, but nevertheless valid, disproof of Beal's equality equation (1). It simply says that Beal's conditions equation (8) are incorrect because z is an integer or irrational number. A full disproof of Beal's equality equation }^{(1), which determines solely that z is an irrational number, is provided by the foregoing bilinear expansion of z equation (4).}

^{}^{}^{Write} **v**^{2/m }^{as an exponential series with terms (ln} **v**^{2}^{/m)}^{n}^{/n! and compare with terms of exponential series e with terms 1}^{/n! }^{}^{which is known to converge. }^{}^{}^{If }^{ln} **v**^{2}^{/m<1}^{}^{}^{}^{}^{}^{}^{ }^{the }*v*^{2/m }series necessarily converges when **v**^{2}^{}^{}< **e**^{m} for all values of v,m and, therefore, from equation (7) **v**^{2}^{}^{}=z^{m}< **e**^{m}, z<e which can occur only when z= 2 when x=y=1. Accordingly, the assumption that equation (7) can be written with z as an integer >2 is refuted and, therefore, z is an irrational number disproving Beal's equality equation (1). QED

^{}

^{1} **The Beal Conjecture and Prize ****http://www.math.unt.edu/~mauldin/beal.html**** **

^{2} **James Constant, Proof of Fermat's Last Theorem ****http://www.coolissues.com/mathematics/Fermat/fermat.htm**** **

**Copyright****©**** 2003 James Constant**

**By same author: ****http://www.coolissues.com/mathematics/sameauthor.htm**