Prime-Generating Formula

(April Fools’!) I came up with this interesting prime-generating formula. It uses the constant \xi and generates the primes in order!

The constant’s approximation.
\xi = 1.603502629914017832315523632362646507807932231768273436867961017532625344 \dots

The formula p_n calculates the n-th prime.
p_n = \lfloor {10^{2 \cdot n} \cdot \sqrt{\xi^3}} \rfloor - \lfloor {10^{2 \cdot (n - 1)} \cdot \sqrt{\xi^3}} \rfloor \cdot 10^2

The first few values for p_n when starting with n=0 are as follows.
p_{0 \text{ to } 7} = \{2, 3, 5, 7, 11, 13, 17, 19, \dots \}

Palindromic Primes

TheOnlinePhotographer has published a post to celebrate 171717 comments and was amused by the number’s symmetry.
A great comment by Lynn pointed out that this number is indeed an interesting number but not symmetrical.
Symmetrical numbers or words also called palindromes are defined as being the same read forwards or backwards. Examples for palindromic words are radar, noon or level. Palindromic numbers are 3, 404 or 172271.

Lynn then went further and checked if 171717 is at least a prime. The number sadly has five distinct prime factors (171717 = 3 \cdot 7 \cdot 13 \cdot 17 \cdot 37).

So Lynn wondered what the next palindromic prime would be.
To answer this question, I wrote this little Python program to check for palindromic primes. The first 120 palindromic primes are shown below.
Based on this list, the smallest palindromic prime larger than 171717 is 1003001.

      3,       5,       7,      11,     101,     131,     151,     181, 
    191,     313,     353,     373,     383,     727,     757,     787, 
    797,     919,     929,   10301,   10501,   10601,   11311,   11411, 
  12421,   12721,   12821,   13331,   13831,   13931,   14341,   14741, 
  15451,   15551,   16061,   16361,   16561,   16661,   17471,   17971, 
  18181,   18481,   19391,   19891,   19991,   30103,   30203,   30403, 
  30703,   30803,   31013,   31513,   32323,   32423,   33533,   34543, 
  34843,   35053,   35153,   35353,   35753,   36263,   36563,   37273, 
  37573,   38083,   38183,   38783,   39293,   70207,   70507,   70607, 
  71317,   71917,   72227,   72727,   73037,   73237,   73637,   74047, 
  74747,   75557,   76367,   76667,   77377,   77477,   77977,   78487, 
  78787,   78887,   79397,   79697,   79997,   90709,   91019,   93139, 
  93239,   93739,   94049,   94349,   94649,   94849,   94949,   95959, 
  96269,   96469,   96769,   97379,   97579,   97879,   98389,   98689, 
1003001, 1008001, 1022201, 1028201, 1035301, 1043401, 1055501, 1062601, ...

Thus it takes 1003001 - 171717 = 831284 more comments to reach the closest palindromic prime.

The sequence of palindromic primes is number A002385 in the On-line Encyclopedia of Integer Sequences (OEIS).


# Python 2.7.7 Code
# Jonathan Frech 23rd of March, 2016

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Haferman Carpet

The Haferman Carpet is a fractal, which kind of looks like a woven carpet. To generate it, you start with a single black pixel and apply in each cycle a set of rules.
In each generation every pixel in the carpet will be replaced by nine pixels according to the rules. A black pixel is represented by a 0, a white one by a 1.

The rules

  • 0 \rightarrow \left( \begin{array}{ccc}1&0&1\\0&1&0\\1&0&1\end{array} \right) \text{and } 1 \rightarrow \left( \begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array} \right)

6 iterations of the Haferman Carpet


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 26th of February, 2016

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