# Which of the following is pseudoprime to the base 2?

## Which of the following is pseudoprime to the base 2?

The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat’s little theorem: 2340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Sarrus numbers, after P. F.

What are strong Pseudoprimes?

A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them “pseudoprimes”.

How many Carmichael numbers are there?

Carmichael numbers are sometimes called “absolute pseudoprimes” and also satisfy Korselt’s criterion. R. D. Carmichael first noted the existence of such numbers in 1910, computed 15 examples, and conjectured that there were infinitely many….Carmichael Number.

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### How do you find pseudoprime?

Given a number N and a base number A. The task is to check whether the number is a Fermat Pseudoprime to the base….Approach: The approach is to check the below conditions:

1. Check if A > 1.
2. Check if N is a composite number.
3. Check if N divides AN-1 – 1.

What are Pseudoprimes used for?

A pseudoprime number is a probable prime number that might actually be a composite number rather than an actual prime. Pseudoprimes are useful in public key cryptography and other aspects of IT.

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.

## How do you find the primality of a number?

The simplest primality test is trial division: given an input number, n, check whether it is evenly divisible by any prime number between 2 and √n (i.e. that the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.

Is Carmichael number 561?

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.

Is 1104 a Carmichael number?

The first Lucas–Carmichael numbers are: 399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, (sequence A006972 in the OEIS)

### Are all Fermat numbers prime?

The only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537. A simple heuristic shows that it is likely that these are the only Fermat primes (though many folks like Eisenstein thought otherwise).

What is the smallest Semiprime number?

Tip − we can also start our algorithm from 6 as the smallest semi-prime number is 6.

Does the number 561 pass the Miller Rabin test?

Therefore 561 does not satisfy the Miller-Rabin test with a = 2, and hence is not prime. Thus our new test finds composite numbers which are missed by Fermat’s test.

## How many pseudoprimes are there to base 2?

There are infinitely many pseudoprimes to a given base (in fact, infinitely many strong pseudoprimes (see Theorem 1 of ) and infinitely many Carmichael numbers ), but they are rather rare. There are only three pseudoprimes to base 2 below 1000, 245 below one million, and only 21853 less than 25·10 9 (see Table 1 of ).

What are the smallest examples of pseudoprimes?

The smallest examples of pseudoprimes for bases 2, 3, 5, and 7 are as follows. 25 = 5.5 is a 7-PRP. It is harder to find examples of composites which are probable primes (or better, strong probable primes) to several bases, but this is always possible [AGP94a]. Let tn be the least composite that is a strong probable prime for the first n primes.

Are only odd numbers pseudoprimes?

Some sources use variations of this definition, for example to allow only odd numbers to be pseudoprimes. An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number .

### What are the best books on pseudoprime statistics?

Pseudoprime Statistics and Tables Pseudoprime Statistics, Tables, and Data (Fermat, Miller-Rabin, Lucas, Fibonacci, Pell, Frobenius, Baillie-PSW) by Dana Jacobsen, 31 March 2020 Limit #PSP-2