Converse
The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers. However, a slightly stronger form of the theorem is true, and is known as Lehmer's theorem. The theorem is as follows:
If there exists an a such that
and for all prime q dividing p − 1
then p is prime.
This theorem forms the basis for the Lucas–Lehmer test, an important primality test.
Read more about this topic: Fermat's Little Theorem
Famous quotes containing the word converse:
“The Anglo-American can indeed cut down, and grub up all this waving forest, and make a stump speech, and vote for Buchanan on its ruins, but he cannot converse with the spirit of the tree he fells, he cannot read the poetry and mythology which retire as he advances. He ignorantly erases mythological tablets in order to print his handbills and town-meeting warrants on them.”
—Henry David Thoreau (1817–1862)
“The eyes of men converse as much as their tongues, with the advantage that the ocular dialect needs no dictionary, but is understood all the world over.”
—Ralph Waldo Emerson (1803–1882)
“Were you to converse with a king, you ought to be as easy and unembarrassed as with your own valet-de chambre; but yet every look, word, and action should imply the utmost respect.... You must wait till you are spoken to; you must receive, not give, the subject of conversation, and you must even take care that the given subject of such conversation do not lead you into any impropriety.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)