BEAL FERMAT PYTHAGORAS

AND HYPERBOLIC GEOMETRY

By James Constant

math@coolissues.com

        In this paper I prove that Hyperbolic geometry cannot be used to prove Pythagora's Theorem, disprove Beal's conjecture, or prove Fermat's Last Theorem. By way of application of these proofs, I show that Wile's proof of Fermat's last theorem and his proof of the Taniyama-Shimura conjecture defy the hyperbolic Pythagorean theorem.

         As a preliminary, expand the hyperbolic cosine as an infinite series


        coshx = 1 +x²/2! + x4/4! + . . . . +x2n/(2n)! + . . . + x2n+2cosh(θx)/(2n+2)!                                                (1)

    where the last term is Lagrange's form of the remainder and note that we can make the approximation as exact as we please for each value of x, since the remainder tends to 0 as n increases.[1]

        Next, consider the Pythagorean theorem for the Hyperbolic plane

        coshz =coshx*coshy                                                                                                                                       (2)

in which, if one replaces cosh by its power series representation, equation (1), and assumes higher order terms for n>2 can be discarded, one obtains as a limit the Pythagorean theorem for the Euclidean plane

        z² = x² + y²                                                                                                                                                        (3)

        However, the assumption that the higher order terms in equation (1) can be discarded requires that terms for n>2 must be significantly less than term x²/2!

        x4/4! + . . . . +x2n/(2n)! + . . . + x2n+2cosh(θx)/(2n+2)!<<x²/2!                                                                     (4)

or by cancelling x²/2! on both sides of equation (4)

        2!x2/4! + . . . . +2!x2n-2/(2n)! + . . . + 2!x2ncosh(θx)/(2n+2)!<<1                                                                  (5)

which states that number x must be some number 2!x2/4!<<1 or x<<23≈3.46. This inequality defines an existence boundary between equations (2) and (3). It does not set a condition for proving either equation.

        Thus, remembering that equation (3) is proven exactly in Euclidean geometry for all rational and irrational numbers z,x,y, a starting from the hyperbolic Pythagorean theorem equation(2), one can only obtain the Pythagorean theorem equation (3) as an approximation for all numbers z<<3.46, x<<3.46, y<<3.46. Therefore, it is impossible to prove exactly Pythagora's theorem equation (3) starting from the hyperbolic Pythagorean theorem equation (2). QED (Pythagoras)

        Now assume Beal's equation can be proven exactly in the Euclidean plane[2]

        zm= xa + yb        z,x,y.m,a,b integers         m,a,b>2                                                                                     (6)

and rewrite equation (6) as a Pythagorean theorem for the Euclidean plane

        z(m/2)2 = x(a/2)2 + y(b/2)2                                                                                                                                    (7)

which corresponds to the hyperbolic Pythagorean theorem

        coshzm/2 =coshxa/2*coshyb/2                                                                                                                           (8)

        Thus, starting from the hyperbolic Pythagorean theorem equation(8), one can only obtain Beal's equation (6) as an approximation for all numbers z<<3.462/m, x<<3.462/a, y<<3.462/b. Therefore, it is impossible to prove exactly Beal's conjecture equation (6) for integers m,a,b>2 starting from the hyperbolic Pythagorean theorem equation (8). QED (Beal )

        Fermat's last theorem is the negation of Beal's equation (6) when m=a=b

        zmxm + ym        z,x,y,m integers         m>2                                                                                                   (9)

        Thus, starting from the hyperbolic Pythagorean theorem equation(8) with m=a=b, one can only obtain Fermat's equality zm=xm + ym as an approximation for all numbers z, x, y<<3.462/mTherefore, it is impossible to prove exactly Fermat's equality zm xm + ymfor integers m>2 starting from the hyperbolic Pythagorean theorem equation (8). QED (Fermat)

        Wile's proof of Fermat's last theorem equation (9) is suspect because it rests on Frey's elliptic equation which Ribet proves is not modular and, therefore, it and  its association with Fermat's equality zm =xm + ym, do not exist.[3] Frey had noted the connection between Fermat's equality zm =xm + ym with elliptic curves. The claimed non modularity of Frey's elliptic equation and non existence of Fermat's equality zm =xm + ym defies the hyperbolic Pythagorean theorem equation (8) with m=a=b which says that Fermat's equality zm =xm + ym exists in hyperbolic cosine form in Hyperbolic space for some numbers z,x,y and exists in approximate Fermat equality form in  Euclidean space for other numbers z,x,y.

        In sum, Wile's proof of the Taniyama-Shimura conjecture says that every elliptic curve is modular and, therefore, Frey's non modular elliptic equation, and non existence of Fermat's equality zm =xm + ym, is a contradiction which proves Fermat's last theorem. Such illogic proof is false because the hyperbolic Pythagorean theorem says that Fermat's equality zm =xm + ym exists in hyperbolic cosine form in Hyperbolic space for some numbers z,x,y and exists in approximate Fermat equality form in Euclidean space for other numbers z,x,y. Frey's non modular elliptic equation contradicts Wile's modularity theorem. We still do not have a satisfactory theory attaching elliptic equations to modular forms.



1    R. Courant, Differential And Integral Calculus, Vol. I p 328, Interscience Publishers, Inc., New York 1937

2    For an actual proof see Beal's Conjecture Disproved at http://www.coolissues.com/mathematics/Beal/beal.htm

3    S. Singh, Fermat's Last Theorem, pp 216-222, Fourth Estate, London 1998

Copyright© 2011 by James Constant

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