# Product of Sequence of Fermat Numbers plus 2

## Theorem

Let $F_n$ denote the $n$th Fermat number.

Then:

 $\displaystyle \forall n \in \Z_{>0}: \ \$ $\displaystyle F_n$ $=$ $\displaystyle \prod_{j \mathop = 0}^{n - 1} F_j + 2$ $\displaystyle$ $=$ $\displaystyle F_0 F_1 \dotsm F_{n - 1} + 2$

### Corollary

Let $F_n$ denote the $n$th Fermat number.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Then:

$F_n \divides F_{n + m} - 2$

where $\divides$ denotes divisibility.

## Proof

The proof proceeds by induction.

For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:

$F_n = \displaystyle \prod_{j \mathop = 0}^{n - 1} F_j + 2$

### Basis for the Induction

$\map P 1$ is the case:

 $\displaystyle F_1$ $=$ $\displaystyle 2^{\paren {2^1} } + 1$ Definition of Fermat Number $\displaystyle$ $=$ $\displaystyle 5$ $\displaystyle$ $=$ $\displaystyle 3 + 2$ $\displaystyle$ $=$ $\displaystyle \paren {2^{\paren {2^0} } + 1} + 2$ $\displaystyle$ $=$ $\displaystyle F_0 + 2$ Definition of Fermat Number $\displaystyle$ $=$ $\displaystyle \displaystyle \prod_{j \mathop = 0}^{1 - 1} F_j + 2$

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

$F_k = \displaystyle \prod_{j \mathop = 0}^{k - 1} F_j + 2$

from which it is to be shown that:

$F_{k + 1} = \displaystyle \prod_{j \mathop = 0}^k F_j + 2$

### Induction Step

This is the induction step:

 $\displaystyle F_{k + 1}$ $=$ $\displaystyle 2^{\paren {2^{k + 1} } } + 1$ Definition of Fermat Number $\displaystyle$ $=$ $\displaystyle 2^{\paren {2 \times 2^k} } + 1$ $\displaystyle$ $=$ $\displaystyle 2^{\paren {2^k} } \times 2^{\paren {2^k} } + 1$ $\displaystyle$ $=$ $\displaystyle \paren {F_k - 1} \times \paren {F_k - 1} + 1$ Definition of Fermat Number $\displaystyle$ $=$ $\displaystyle \paren {\paren {\prod_{j \mathop = 0}^{k - 1} F_j + 2} - 1} \paren {F_k - 1} + 1$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle \paren {\prod_{j \mathop = 0}^{k - 1} F_j} F_k + F_k - \prod_{j \mathop = 0}^{k - 1} F_j - 1 + 1$ multiplying out $\displaystyle$ $=$ $\displaystyle \prod_{j \mathop = 0}^k F_j + F_k - \paren {F_k - 2}$ Induction Hypothesis, and simplifying $\displaystyle$ $=$ $\displaystyle \prod_{j \mathop = 0}^k F_j + 2$ simplifying

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\forall n \in \Z_{>0}: F_n = \displaystyle \prod_{j \mathop = 0}^{n - 1} F_j + 2$

$\blacksquare$