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
Famous quotes containing the words examples of, examples, small and/or radical:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“And still they gazed, and still the wonder grew,
That one small head could carry all he knew.”
—Oliver Goldsmith (1728–1774)
“Every genuine boy is a rebel and an anarch. If he were allowed to develop according to his own instincts, his own inclinations, society would undergo such a radical transformation as to make the adult revolutionary cower and cringe.”
—Henry Miller (1891–1980)