**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**