The Goldbach conjecture.

Proof the Riemann-Hypothesis and Goldbach’s conjecture

Sort these conjectures into "always true", "sometimes true" and "never true".always true sometimes true never trueA prime number has an even number of factors. The sum of two consecutive numbers is a multiple of 2. A square number has an even number of factors.

The HCF of the smallest prime number and the smallest composite number is

The Collatz Conjecture, also known as the 3n + 1 problem, is a conjecture in number theory, first proposed by Lothar Collatz in 1937. The conjecture is as follows:Start with any positive integer 𝑛n. If 𝑛n is even, divide it by 2. If 𝑛n is odd, triple it and add 1. Repeat this process indefinitely. The conjecture states that no matter what value of 𝑛n you start with, you will always eventually reach the cycle 4,2,14,2,1.For example, starting with 𝑛=7n=7, the sequence is 7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,17,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1.The Collatz Conjecture remains unproven, despite extensive computational verification for extremely large numbers. It is considered one of the most perplexing and enduring unsolved problems in mathematics. So, if you're up for a real challenge, attempting to prove or disprove the Collatz Conjecture could be an exciting purs

\begin{problem} For $n\in\N$, let $\sigma_k(n)$ be defined as the sum of the $k$th power of the positive divisors of $n$, i.e.: \[ \sigma_k(n) = \sum\limits_{d|n} d^k. \] Let $p$ be prime and $e\in \N_0$. Through experimentation, develop conjectures for the following: \begin{enumerate} \item $\sigma_0(p^e)$ \item $\sigma_1(p)$ \item $\sigma_1(p^e)$ \end{enumerate} Then, choose one of the above to prove.\end{problem}\begin{proof}\end{proof}