World!Of
Numbers

WON plate
146 |

[ 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```

A000146 Prime Curios! Prime Puzzle
Wikipedia 146 Le nombre 146
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