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Disproving the ABC Conjecture
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.
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 a and b. If m=2 then d=ijk and c=i2, and thus d>s or d<s, and equation (1) reduces to i2=a2+b2, an impossibility for all
values of a and b. Accordingly, 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, jn, ko. 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.
Copyright © 2013 James Constant
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