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)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“There were none of the small deer up there; they are more common about the settlements. One ran into the city of Bangor two years before, and jumped through a window of costly plate glass, and then into a mirror, where it thought it recognized one of its kind.... This the inhabitants speak of as the deer that went a-shopping.”
—Henry David Thoreau (1817–1862)
“Radical Chic, after all, is only radical in Style; in its heart it is part of Society and its traditions—Politics, like Rock, Pop, and Camp, has its uses.”
—Tom Wolfe (b. 1931)