Triskaidekaphilia (Part I)

As Discrete Connections will turn soon 13 (for those frail at heart, please do stay away, you may never know what may happen :P), I decided to start another thread concerning, as you might have guessed, the arithmetic properties of number 13 (and other prime numbers to that respect), some old, some fresh, but altogether hopefully increasingly interesting. Wherever possible open questions will be raised, and extensions for other prime numbers will be mentioned (please forgive me if I fail in finding references to some of the results to follow, it is not intentional but merely a proof I'm not nor ever will be a know-it-all).

1. 13 is a prime number.

This should have not come as a surprise, but I ought have mentioned it as it one of the basic properties of this number.

2. 13 is a prime that can be written as a sum of squares of natural numbers.

Indeed:

[; 13 = 2^2 + 3^2;]

It is not the first prime having this property:

[;2 = 1^2 + 1^2, \; 5 = 1^2 + 2^2;]

but it's the first for which the squared numbers are also primes. Are there an infinity of such primes? Yes as this is a consequence the renowned sum of squares theorem attributed to Fermat stating that this property holds for a prime if and only if that prime is congruent to 1 modulo 4. Since there are an infinity of those, thanks to Dirichlet's theorem stating that any arithmetic progression having relatively prime first term and ratio contain an infinity of primes (1 and 4 are such numbers), the conclusion follows naturally.

3. 13 is a prime that divides the sum of squares of all the previous prime numbers.

Indeed:

[;2^2 + 3^2 + 5^2 + 7^2 + 11^2 = 208 = 13 \cdot 16;]

Here 13 is definitely the first prime with this property. Is it the only one, are there infinitely more such primes? Hard to answer (in my opinion very unlikely).

4. 13 is a prime belonging to the Fibonacci sequence.

Not the first (2,3 and 5 are also), not the last, just one of the many (not yet known if there are infinitely many, but there are some big ones).

5. 13 is a prime with the property that:

[;\tau_2(p) = p-2;]

where the notation follows Discrete connections (the euclidean summation peak).

Here there are as many primes as in 4 since:

[;\tau_2(n) = n-2 \Leftrightarrow \exists m \in \mathbf{N} \; such \; that \; n = F_{m} \ge 2;]