The erroneous proof I hear most often is: Suppose \(P\) is a finite set that contains all the primes, then \(p^* = 1 + \prod_{p \in P} p\) is prime. Indeed, the flaw is that \(p^*\) is not necessarily prime but rather must be a multiple of some prime not in \(P\).

$$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} $$

The Fibonacci numbers are defined as \(f_0 = 0,\ f_1 = 1\) and, for \(i \ge 2,\ f_i = f_{i-1} + f_{i-2}\). Here is the beginning of the Fibonacci sequence:

\(0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots\)

We generalize the definition above by changing the two initial values, for example with \(f_0 = 4,\ f_1 = 6\) we obtain

\(4, 6, 10, 16, 26, 42, \ldots\)