Examples of Triples With Small Radical
The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples a, b, c with rad(abc) < c. For instance, such a triple may be taken as
- a = 1
- b = 26n − 1
- c = 26n
As a and c together contribute only a factor of two to the radical, while b is divisible by 9, rad(abc) < 2c/3 for these examples. By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c may be made arbitrarily small. Specifically, replacing 6n by p(p-1)n for an arbitrary prime p will make b divisible by p2, because 2p(p-1) ≡ 1 (mod p2) and 2p(p-1) - 1 will be a factor of b.
A list of the highest quality triples (triples with a particularly small radical relative to c) is given below; the highest quality of these, with quality 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137):
- a = 2
- b = 310 109 = 6,436,341
- c = 235 = 6,436,343
- rad(abc) = 15042
Read more about this topic: abc Conjecture
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