Sample Consequence
It is no accident that
or equivalently,
is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that
is a unique factorization domain.
Here satisfies α² = α − 41. In general, S denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.
Alternatively,
an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.
Read more about this topic: Complex Multiplication
Famous quotes containing the words sample and/or consequence:
“All that a city will ever allow you is an angle on it—an oblique, indirect sample of what it contains, or what passes through it; a point of view.”
—Peter Conrad (b. 1948)
“Cunning is neither the consequence of sense, nor does it give sense. A proof that it is not sense, is that cunning people never imagine that others can see through them. It is the consequence of weakness.”
—Horace Walpole (1717–1797)