. . . . . . . . x,y,z,m,a,b positive
integers . . . . . . m,a,b>2 BEAL'S CONJECTURE DISPROVED
James Constant
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 conjecture1, 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 expansion2. 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 xa/m.
When divergent, the parenthesis term series and thus z
in
equation (4) have no values.
3. For yb/xa=+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^kk
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 xa=yb,x=2^kk
integer, ka>=m-1 and
ka>=b.
See also Beal's
Conjecture And The Problem of Multiplying Irrational Number's at
Mirabilis Disproof of Beal's Conjecture
Write the right hand side of equation (1) as
(5) x2a/2 + y2b/2 x,y,a,b integers a,b>2
which, by Pythagora's theorem, is a real number squared, say
(6) v2=xa + yb v2 >2 or = 2 when x=y=1
in which v2 is an integer and v is an integer or irrational number. Equality equation (1) is obtained by assuming
(7) v2=zm m > 2 positive integer
in which zm 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 v2/m as an exponential series with terms (ln v2/m)n/n! and compare with terms of exponential series e with terms 1/n! which is known to converge. If ln v2/m<1 the v2/m series necessarily converges when v2< em for all values of v,m and, therefore, from equation (7) v2=zm< em, 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