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Disproving the ABC Conjecture

James Constant

math@coolissues.com

The abc conjecture purports to tell us something about Diophantine equations in terms of inequality d=rad(abc)<c in which a, b, and c are coprime positive integers such that c=a+b  and  d=rad(abc) is the product of  primes in product abc

without exponents. Usually d=rad(abc)>c. However, if c=im, a=jn, b=ko with positive integers m,n,o, then d=rad(abc)<c becomes possible since d=ijk could be smaller than c=im

To disprove the conjecture, write equation c=a+b as a Pythagorean form with  c=im, a=jn, b=ko.

(im/2)2=(jn/2)2+(ko/2)2                                           (1)

invoke Euclid's formulas for triplets  and solve for i,j,k

i = (r2  +  s2)2/m           j = (2rs)2/n            k = (r2  -  s2)2/o    (2)

in which r,s are integers.

Equation (1) states that, since im=jn+ ko is not known beforehand, the existence of the equality for prime triplets i,j,k must be proven for positive integers m,n,o.

Since i,j,k are integers, equations (2) are impossibilities unless m,n,o are integers 1 or 2. Note that equation (1)  in form is Beal's equality when m,n,o>2 are different integers, in form is Fermat's equality when m=n=o>2, in form is

Pythagora's equality when m=n=o=2, and in form is i = j + k when m=n=o=1. Accordingly, triplets i,j,k do not  exist when m,n,o>2 or when m=n=o>2.

If m=1 then d=ijk and c=i and thus d>c, and equation (1) reduces to  i=a2+b2, an impossibility for all values of and bIf m=2  then d=ijk and c=i2, and thus d>s or d<s, and equation (1) reduces to  i2=a2+b2an impossibility for all

values of and bAccordingly, triplets i,j,k do not exist for all values of a and b when m=1 or m=2.

More generally, write equation c = a + b as a Pythagorean form with c=Πim, a=Πjn, b=Πko, in which products Π run through the different values of im, jnko. Equations (2) become

c=Πim = (r2  +  s2)2          a=Πjn = (2rs)2           b=Πko(r2  -  s2)2      (3)

Equation (3) states that, since c=a+b is known beforehand from arithmetic, triplets a,b,c exist for all numbers b,a. However, nothing can be said of individual primes. The abc conjecture is disproved because it tells us nothing we did not know

about Diophantine equations.  Inequality d=rad(abc)<c  plays no role in finding solutions to Diophantine equations.

 http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.html