 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 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

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

thus proving FLT

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

Disproof of Beal's Conjecture

When a=b=m, Beal's equation (1) becomes Fermat's equation (2). Clearly, Fermat's equation (2) is a special case of Beal's equation (1). The same procedure used in Fermat's equation (2) can be used to show that z cannot be an integer in Beal's equation (1). Start by rearranging Beal's equation (1)

(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.

Mirabilis Disproof of Beal's Conjecture

because z is an integer or irrational number. 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