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:
““You are old, Father William,” the young man cried,
“And life must be hastening away;
You are cheerful, and love to converse upon death:
Now tell me the reason, I pray.”
“I am cheerful, young man,” Father William replied;
“Let the cause thy attention engage;
In the days of my youth I remembered my God,
And He hath not forgotten my age.””
—Robert Southey (1774–1843)
“The American doctrinaire is the converse of the American demagogue, and, in this way, is scarcely less injurious to the public. The first deals in poetry, the last in cant. He is as much a visionary on one side, as the extreme theoretical democrat is a visionary on the other.”
—James Fenimore Cooper (1789–1851)
“I lately met with an old volume from a London bookshop, containing the Greek Minor Poets, and it was a pleasure to read once more only the words Orpheus, Linus, Musæus,—those faint poetic sounds and echoes of a name, dying away on the ears of us modern men; and those hardly more substantial sounds, Mimnermus, Ibycus, Alcæus, Stesichorus, Menander. They lived not in vain. We can converse with these bodiless fames without reserve or personality.”
—Henry David Thoreau (1817–1862)