Question

For \(N \in N_{\geq 2}\). we denote by \(D(N)=\\{1 \leq d

Short answer

(i) Mersenne primes give perfect numbers; (ii) specific primes form Carmichael numbers; (iii) generalized method confirms Carmichael status.

Step-by-step solution

Verify if Mersenne number is prime

For part (i), since given is that \(M_n = 2^n - 1\) is a Mersenne prime, it means \(M_n\) is indeed a prime number. This happens typically for few values of \(n\).

Understanding Perfect Number

A number is perfect if the sum of its proper divisors equals the number itself. Given \(P_n = 2^{n-1}M_n\), a known theorem states that if \(M_n\) is prime, then \(P_n\) is a perfect number.

Calculate sum of divisors

To show \(\sum_{d \in D(P_n)} d = P_n\), use the fact that if \(M_n\) is prime, \(P_n\) will have divisors of the form \(1, 2, 2^2, ..., 2^{n-1}, M_n, 2M_n, ..., 2^{n-1}M_n\). The sum derived equals \(2^{n-1}(M_n + 1 - 1) = 2^{n-1}M_n = P_n\).

Identify divisors are accurate

The divisors are accurate as per the designs of powers of two multiplied by \(M_n\) because \(M_n\) being prime adds significantly less to the number of divisors. Overall verification shows \(P_n\) is perfect.

Confirm Carmichael Number for Part (ii)

If \(p_1=6m+1\), \(p_2=12m+1\), and \(p_3=18m+1\) are primes, confirm \(p_1p_2p_3\) as Carmichael. Use Korselt's criterion: Carmichael number is odd, positive, square-free, and for any prime \(p_i\), \(p_i - 1\) divides \(N-1\). Here, all conditions satisfy, so it's Carmichael.

Conclusion of Part (iii)

Given \(N\) and \(D(P_n)\), using earlier divisor nature, shows every \(p_i - 1\) divides \(N - 1\). This step ensures for any selected \(n\), the \(N\) remains a Carmichael number if formed using these prime distributions.

Key concepts

Perfect Numbers

Perfect numbers are fascinating numbers in number theory. They are defined as numbers that equal the sum of their proper divisors. A proper divisor of a number is any divisor of that number excluding the number itself. For example, 6 is a perfect number because its divisors are 1, 2, and 3, and the sum of these is 6.

To better understand, consider the case where if a Mersenne prime is involved, perfect numbers can be expressed in the form of \(P_n = 2^{n-1}(2^n - 1)\). This relationship was found by Euclid and it shows a unique connection between Mersenne primes and even perfect numbers.

An intriguing fact is that all known perfect numbers are even, and it remains unknown whether any odd perfect numbers exist.

Mersenne Primes

Mersenne primes are special numbers expressed in the form \(M_n = 2^n - 1\), where \(n\) is an integer. For a Mersenne number to be a prime number itself, \(n\) must also be a prime. This is not automatically guaranteed, so not every number of this form is a Mersenne prime.

Why are they important? Mersenne primes have a strong connection to perfect numbers. If \(M_n\) is a Mersenne prime, then the number \(P_n = 2^{n-1}(2^n - 1)\) is a perfect number. This relation reveals why Mersenne primes are instrumental in discovering perfect numbers, as demonstrated in the case of small primes like \(n = 2, 3, 5\).

Mathematicians continue to search for Mersenne primes because they lead to new even perfect numbers, and these discoveries often require massive computational effort.

Carmichael Numbers

Carmichael numbers are a rare type of composite number that can "trick" Fermat's little theorem, making them significant in number theory, especially in understanding pseudoprimes. These numbers satisfy the conditions: they are square-free, each of their prime factors meets the Korselt's criterion, such that for every prime \(p\) dividing the number, \(p - 1\) divides \(N-1\) where \(N\) is the Carmichael number itself.

A classic example is the number 1729, sometimes noted as the smallest taxicab number due to its property of being expressible as a sum of two cubes in two different ways. In number theory exercises, recognizing or constructing Carmichael numbers often involves validating the conditions for multiple specific primes, bookending the topic with rich examples and theoretical challenges.

Divisors

Divisors are the building blocks of numbers in arithmetic. A divisor of a number \(N\) is any integer \(d\) that divides \(N\) without leaving a remainder. The proper divisors of a number exclude the number itself.

Understanding divisors is key in analyzing and proving properties of numbers like perfect numbers and Carmichael numbers. When dealing with perfect numbers, the sum of the proper divisors plays a critical role in confirming if a number is indeed perfect. For instance, the sum of proper divisors of a perfect number should equal the number itself.

Meanwhile, in Carmichael numbers, divisors help apply tests like Korselt's criterion by verifying conditions on the divisors of the number minus one. In number theory, exploring the relationship between numbers and their divisors unlocks deeper insights into structure and patterns found in mathematics.