Source: Imgur at http://imgur.com/I3lp1CT In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e. not prime) the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them which is equal to that prime.[1] This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. The sieve of Eratosthenes is one of the most efficient ways to find all of the smaller primes (below 10 million or so). It is named after Eratosthenes of Cyrene, a Greek mathematician; although none of his works have survived, the sieve was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus. The sieve may be used to find primes in arithmetic progressions. This is from Calculus Humor's wiki at http://calculushumor.wikia.com/wiki/Sieve_of_Eratosthenes! The Count Dracula awakes on night 0 and his evil afterlife begins. On night 1, Dracula starts biting people. Every bitten human becomes a vampire and bites people as well. Let $v(n)$ be the vampire population after $n^\text{th}$ night. Investigate the function $v(n)$ in the following cases: a) Every vampire bites one human every night starting the first night after the initiation. Write down a recurrent and an explicit formula for $v(n)$. b) Every vampire bites one human every night starting the second night after the initiation. Write down a recurrent formula for $v(n)$. Prove that \[v\left ( n \right )=\frac{\left ( \frac{1+\sqrt{5}}{2} \right )^{n+2}\left ( \frac{1\sqrt{5}}{2} \right )^{n+2}}{\sqrt{5}}\] c) Every vampire bites $m$ humans on the $n^\text{th}$ night after the initiation. Write down a recurrent and an explicit formula for $v(n)$.

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