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Day #10: ABC Conjecture

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Shinichi Mochizuki

Q: Can you see the equivalence of the three following statements of the ABC conjecture??…

ABC Conjecture I. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:

c>\operatorname {rad} (abc)^{1+\varepsilon }.

ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:

  {\displaystyle c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.}

ABC Conjecture III. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

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Day #5: From Pythagoras to the Mathematics of the 3rd Millennium

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Q: Could you give at least 10 more proofs of the Pythagorean Theorem, and state how it is related to todays most actual mathematics and the most famous mathematical problem of all times??…

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