Has never been such a large-scale attack on Diophantine equations.
In this hypothesis, the number of results axceeded oll expectations.
Hypothesis. For evere r>0 and for oll triples (a,b,c) of coprime positive integers, with a+b=c, the inequality c<=(rad(c))r or c<(rad(ab))r holds.
if 0<r<=1 then hypothesis is wrong for infinitely many triples (a,b,c) of positive coprime integers;
if 1<r<2 then hypothesis is wrong for only finitely many triples (a,b,c) of positive coprime integers;
if r>=2 then hypothesis is true for oll triples (a,b,c) of positive coprime integers.
Or else
Ar={(a,b,c)|a,b,c cN, a+b=c, (a,b)=1, c>(rad(c))r, c>=(rad(ab))r}
Conjecture. if 0<r<=1 then card Ar is infinite;
if 1<r<2 then card Ar is finite;
if r>=2 then card Ar is zero.
Emzari Papava
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