Explanation of The Wieferich Property
The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation 2p − 1 ≡ 1 (mod p2). From the definition of the congruence relation on integers, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime p satisfies this congruence, this prime divides the Fermat quotient . The following are two illustrative examples using the primes 11 and 1093:
For p = 11, we get which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For p = 1093, we get or 530585362....3096656895 (320 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.
Read more about this topic: Wieferich Prime
Famous quotes containing the words explanation of the, explanation of, explanation and/or property:
“Herein is the explanation of the analogies, which exist in all the arts. They are the re-appearance of one mind, working in many materials to many temporary ends. Raphael paints wisdom, Handel sings it, Phidias carves it, Shakspeare writes it, Wren builds it, Columbus sails it, Luther preaches it, Washington arms it, Watt mechanizes it. Painting was called “silent poetry,” and poetry “speaking painting.” The laws of each art are convertible into the laws of every other.”
—Ralph Waldo Emerson (1803–1882)
“To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.... It must involve throughout a resolute rejection of the demand for an explanation of the regularities in the observable course of nature, by means of truths concerning a reality beyond what is actual and observable, as a demand which plays no role in the scientific enterprise.”
—Bas Van Fraassen (b. 1941)
“Are cans constitutionally iffy? Whenever, that is, we say that we can do something, or could do something, or could have done something, is there an if in the offing—suppressed, it may be, but due nevertheless to appear when we set out our sentence in full or when we give an explanation of its meaning?”
—J.L. (John Langshaw)
“For wisdom is the property of the dead,
A something incompatible with life; and power,
Like everything that has the stain of blood,
A property of the living; but no stain
Can come upon the visage of the moon
When it has looked in glory from a cloud.”
—William Butler Yeats (1865–1939)