Triskaidekaphilia (Part II)

6. 13 is a prime such that for any digits a,b > 0 and any integers n,m > 1 one can build an integer divisible by it having exactly n digits equal to a, exactly m digits equal to b and exactly 2 digits equal to 0. Moreover this m+n+2 digit long number is non-trivial, in that none of the two 0 digits is located at the end of the number.

Indeed this is true since:
\(13 \mid 10\underbrace{1 \ldots 1}_{k \; times \; 1}01, \forall k \in \mathbf{N}\)

One other prime with this property is 7. This can easily e extended to any number of digits.

7. 13 is a prime with the property that:

\( \forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid \overline{ca} \)

This is equivalent to the fact that the following equation system in a, b and c:

\( \left\{ \begin{array}{ccccccc} 10 \cdot a & + & 1 \cdot b & + & 0 \cdot c & = & 0 \\ 0 \cdot a & + & 10 \cdot b & + & 1 \cdot c & = & 0 \\ 1 \cdot a & + & 0 \cdot b & + & 10 \cdot c & = & 0 \end{array} \right. \)

has non-trivial solutions modulo p, hence for its discriminant we have that:

\( \left| \begin{array}{ccc} 10 & 1 & 0 \\ 0 & 10 & 1 \\ 1 & 0 & 10 \end{array}\right| = 1001 \equiv 0 \mod{p} \)

Therefore the primes with this property are 7, 11 and 13.

8. 13 is a prime with the property that:

\(\forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid a + b + c\)

9. 13 is a prime with the property that:

\(\forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid a^2 + b^2 + c^2\)

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;]

Discrete connections (part XII)

First a small stop to explain this sudden "gold rush" about finite continued fractions and how they impact upon the subject at stake.

1. As mentioned previously, they are related to the euclidean fractal by its construction. Indeed, consider:

[; the \; (p_{s}-1)^{th} \; block \; of \; \ldots \; of \; the \; (p_{1}-1)^{th} \; block \; of ;]
[; the \; euclidean \; fractal \; defined \; by \; u=0, v=i, w=1+i, z=1 ;]

then its base endpoints' real parts are:

[;\cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s}}}} \; and \; \cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + 1}}}};]

2. As you might already know, any positive rational number can be expressed as a continued fraction whose components are, in a top-down representation, the successive quotients obtained through Euclid's gcd algorithm applied to the numerator and denominator. Moreover, there is only one such canonical representation, i.e. if a sub-unitary rational number k/n is equal to a canonical finite continued fraction then we have all the steps of Euclid's gcd algorithm (we have all the quotients) for n and k.

Further more, if we strictly bound a positive sub-unitary rational number k/n to values such as described in 1, then Euclid's gcd algorithm for n and k has at least s steps, for which the quotients are given by the continued fractions defining the boundaries.

Discrete connections (part XI)

This part can be better classified as a "bigger picture" overview. We've seen in the past articles several strong hints concerning the global properties of the euclidean summation but no word yet about local ones. Here we try to give some of these local properties a go, while raising subtle bridges between the euclidean summation and the euclidean fractal.

As accustomed, we will define some derivatives of the euclidean summation peak and euclidean summation positional peak:

[; \tau_2^{p}(n) = \left\{ \begin{array}{l} 0, \; if \; I_n^{p} = \emptyset \\ \max\{ \tau_1(n,k) \; | \; k \in I_{n}^{p} \}, \; otherwise \end{array} \right. ;]

[; \tau_3^{p}(n) = \left\{ \begin{array}{l} 0, \; if \; I_{n}^{p} = \emptyset \\ \max\{ k \in I_{n}^{p} \; | \; \tau_1(n,k) = \tau_2^{p}(n) \}, \; otherwise \end{array} \right. ;]

where:

[; p \in \mathbf{N}, \; p \ge 1 ;]

[; I_{n}^{p} = \{k \in \mathbf{N} \; | \; \frac{n}{p+1} \le k \le \frac{n}{p} \} ;]

The target conjectures for these derivatives are:

[; \lim_{n \rightarrow \infty} \frac{\tau_2^{p}(n)}{n} = \frac{\varphi}{p + \phi} ;]

[; \lim_{n \rightarrow \infty} \frac{\tau_3^{p}(n)}{n} = \frac{1}{p+\phi} ;]

For p=1 these conjectures translate to the result proven for the euclidean summation peak and respectively to the conjecture for the euclidean summation positional peak.

We can go even deeper with this construction, and involve yet another classic: continued fractions (and thus tighten the links with the euclidean fractal). Indeed:

[; \forall s \in \mathbf{N}, \; s > 0 \; and \; \forall w = (p_1, \ldots, p_s) \in (\mathbf{N} \setminus \{0\})^{s} ;]

let us define:

[; \tau_2^{w}(n) = \left\{ \begin{array}{l} 0, \; if \; I_n^{w} = \emptyset \\ \max\{ \tau_1(n,k) \; | \; k \in I_{n}^{w} \}, \; otherwise \end{array} \right. ;]
[; \tau_3^{w}(n) = \left\{ \begin{array}{l} 0, \; if \; I_{n}^{w} = \emptyset \\ \max\{ k \in I_{n}^{w} \; | \; \tau_1(n,k) = \tau_2^{w}(n) \}, \; otherwise \end{array} \right. ;]

where

[; I_{n}^{w} = \{k \in \mathbf{N} \; | \; \cfrac{n}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + \epsilon(s)}}}} \le k \le \cfrac{n}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + 1 - \epsilon(s)}}}} \} ;]

and:

[;\epsilon(s) = \left\{ \begin{array}{l} 0, \; if \; s \; even \\ 1, \; otherwise \end{array} \right. ;]

The conjectured results for these generalized derivatives are:

[; \lim_{n \rightarrow \infty} \frac{\tau_3^{w}(n)}{n} = \cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + \phi }}} ;]

Next: To be continued ...