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:
“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)
“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)
“Thirty-five years ago, when I was a college student, people wrote letters. The businessman who read, the lawyer who traveled; the dressmaker in evening school, my unhappy mother, our expectant neighbor: all conducted an often large and varied correspondence. It was the accustomed way of ordinarily educated people to occupy the world beyond their own small and immediate lives.”
—Vivian Gornick (b. 1935)
“Any critic is entitled to wrong judgments, of course. But certain lapses of judgment indicate the radical failure of an entire sensibility.”
—Susan Sontag (b. 1933)