## Factorization

Playing around with prime numbers, I created this simple factorization program.
The interesting thing about prime factors is that they are unique. There can only be one way to multiply prime numbers to get $n$ where $n \in \mathbb{N}$ and $n \geq 2$ (excluding the commutative property).
For example, $2 \cdot 3 \cdot 7 = 42$ and that is the only way to multiply prime numbers to get to $42$.

# Python 2.7.7 Code
# Jonathan Frech 8th of April, 2016

## 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