[ *February 16, 2003* ]

A threefold (probable) prime search

Find the smallest prime(s) composed of the successive concatenation

of the prime factors (counting multiplicity) of the composites.

This integer sequence starts with 4 (see A002808).

[4] | = 2 * 2 |

[6] | = 2 * 3 |

[8] | = 2 * 2 * 2 |

[9] | = 3 * 3 |

[10] | = 2 * 5 |

[12] | = 2 * 2 * 3 |

Thus the (probable) prime(s) we are looking for begins like this

(2 * 2)(2 * 3)(2 * 2 * 2)(3 * 3)(2 * 5)(2 * 2 * 3)...

or in its pure format

22232223325223...

Note that when a next composite is added one has

to append ALL its prime factors to the string.

Secondly find similar prime(s) (>23) but this time include the

primes themselves in the composite prime factors string.

The sequence starts with 2 (see A000027 but without the unity 1).

[2] | = prime |

[3] | = prime |

[4] | = 2 * 2 |

[5] | = prime |

[6] | = 2 * 3 |

[7] | = prime |

[8] | = 2 * 2 * 2 |

[9] | = 3 * 3 |

[10] | = 2 * 5 |

[11] | = prime |

[12] | = 2 * 2 * 3 |

Thus the (probable) prime(s) we are looking for begins like this

(2)(3)(2 * 2)(5)(2 * 3)(7)(2 * 2 * 2)(3 * 3)(2 * 5)(11)(2 * 2 * 3)...

or in its pure format

23225237222332511223...

Thirdly the exact exercice as above but with unity included.

[1] | = unity |

[2] | = prime |

[3] | = prime |

[4] | = 2 * 2 |

[5] | = prime |

[6] | = 2 * 3 |

Thus the (probable) primes we are looking for begins like this

(1)(2)(3)(2 * 2)(5)(2 * 3)...

Here are already the first three solutions

From 1 to 6

12322523

From 1 to 27

12322523... ...23222355213333

From 1 to 53

12322523... ... 472222377255317221353

So what comes after 6, 27 and 53 ?

[ *February 18, 2003* ]

Jeff Heleen believes to have found a prime for part 1.

Using the composite numbers from 4 to **555** (palindromic!)

gives a probable prime.

He has it running on Primo2 and should be finished

by the time he gets home (thursday nite).

" Nothing yet on parts 2 and 3 but still looking. "

[ *February 25, 2003* ]

Jeff's "Primo 2.0.0 - beta 3" certificate and validation

for the number of part 1 is now available.

[PRIMO - Primality Certificate]
Version=2.0.0 - beta 3
Format=2
ID=B288601551318
Created=02/24/2003 06:12:32 AM
TestCount=291
Status=Candidate certified **prime**
[Candidate]
File=C:\Program Files\Primo200\Work\Arnault.in
N$=C1934B4458... ...436FD36221
HexadecimalSize=1570
DecimalSize=1891
BinarySize=6280
[Running Times]
Initialization=12.94s
1stPhase=27h 28mn 19s
2ndPhase=2h 44mn 2s
Total=30h 12mn 35s |