Sage-Tips

Is 341 a Carmichael number?

The smallest base-2 Fermat pseudoprime is 341. An integer x that is a Fermat pseudoprime for all values of a that are coprime to x is called a Carmichael number.

Why is 561 a Carmichael number?

3. Hence, 561 is a Carmichael number, because it is composite and b560 ≡ (b80)7 ≡ 1 mod 561 for all b relatively prime to 561. for all b relatively prime to 1105. Hence, 1105 is also a Carmichael number.

What is absolute pseudoprime?

Definition. An absolute pseudoprime (or a Carmichael number) is a composite number n > 1 such that an ≡ a (mod n) for every integer a. (4) Show that if k is an integer, then one of the two consecutive numbers k2010 − 1 and k2010 is divisible by 31.

Is 1729 a Carmichael number?

The third Carmichael number (1729) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes (of positive numbers) in two different ways.

Is 341 a absolute pseudoprime?

pseudoprime, a composite, or nonprime, number n that fulfills a mathematical condition that most other composite numbers fail. However, 341 = 11 × 31, so it is a composite number. Thus, 341 is a Fermat pseudoprime to the base 2 (and is the smallest Fermat pseudoprime).

What are the factors of 341?

Prime Factors of 341 : 11 * 31.

Is 62745 a Carmichael number?

The Fermat probable primality test will fail to show a Carmichael number is composite until we run across one of its factors! The Carmichael numbers under 100,000 are. 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361.

Is 1105 a Carmichael number?

1105 is a Carmichael number: ∀a∈Z:a⊥1105:a1105≡a(mod1105) while 1105 is composite.

What does it mean for an integer n to be a Pseudoprime to a base A?

In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and. (where mod refers to the modulo operation). The motivation for this definition is the fact that all prime numbers p satisfy the above equation which can be deduced from Fermat’s little theorem.

How do I find my Carmichael number?

A composite integer n is a Carmichael number if and only if an ≡ a mod n for all a ∈ Z. Proof. If an ≡ a mod n for all a ∈ Z, then when (a, n) = 1 we can cancel a from both sides and get an-1 ≡ 1 mod n, so n is a Carmichael number since it is composite.

How do I prove my Carmichael number?

What does it mean for an integer n to be a pseudoprime to a base A?

Is the number 341 prime or composite?

However, 341 = 11 × 31, so it is a composite number. Thus, 341 is a Fermat pseudoprime to the base 2 (and is the smallest Fermat pseudoprime). Thus, Fermat’s primality test is a necessary but not sufficient test for primality. As with many of Fermat’s theorems, no proof by him is known to exist.

Is 341 a Fermat pseudoprime to base 2?

Thus, 341 is a Fermat pseudoprime to the base 2 (and is the smallest Fermat pseudoprime). Thus, Fermat’s primality test is a necessary but not sufficient test for primality.

What is a pseudoprime in math?

Pseudoprime, a composite, or nonprime, number that fulfills a mathematical condition that most other composite numbers fail. The best-known of these are the Fermat pseudoprimes that fulfill Fermat’s Little Theorem, in which a number n such that it divides exactly a^n – a for some integer a.

Who discovered Fermat pseudoprime?

The first known proof of this theorem was published by Swiss mathematician Leonhard Eulerin 1749. There exist some numbers, such as 561 and 1,729, that are Fermat pseudoprime to any base with which they are relatively prime. These are known as Carmichael numbers after their discovery in 1909 by American mathematician Robert D. Carmichael.