Since when was 4096 prime?

[Gordon]

In the recently cleared exponents list is this little entry

Jacques MOLNE
mon_4
4096
F-ECM
2015-07-03 00:08
5.5
1.0369
45477879701734570611058964078361695337745924097

Thought all exponents had to be prime which clearly an even number isn't (apart from 2 to head off RDS)

[axn]

F-ECM stands for Fermat-ECM. This is a result of ECM run on 2^4096+1. Incidentally, the factor found is a composite one consisting of some of the previously-known factors for that number. As for "should it be there in the cleared exponents list", I'll let the project admins make that decision.

[petrw1]

Originally Posted by axn

F-ECM stands for Fermat-ECM. This is a result of ECM run on 2^4096+1. Incidentally, the factor found is a composite one consisting of some of the previously-known factors for that number. As for "should it be there in the cleared exponents list", I'll let the project admins make that decision.

JM reports these same factors over and over and over as indicated here with the 113 successes.
I suspect NONE of these 113 are NEW.

Top ECM on Fermat numbers producers for the last year.

                                  Total                     |      Rank Change       | Overtake date based on
 Rank  Member Name              GHz-Days Attempts Successes |90day 30day 7 day 1 day | 90 days 30 days  7 days
------ -------------------- ------------ -------- --------- |----- ----- ----- ----- | ------- ------- -------
     1 David Bessell           43850.855     6261         0 |                        |                 >999dy
     2 Jacques MOLNE           33930.562     2831       113 |                        |

And here:

PrimeNet Most Recent Cleared Not-Prime
This report updates every hour on the hour.  This report produced 2015-07-03 15:00 UTC
Member Name  Computer Name  Exponent  Type  UTC Time Received  Days  GHz-days  Result
Jacques MOLNE	mon_4	4096	F-ECM	2015-07-03 10:05	5.9	1.0369	45477879701734570611058964078361695337745924097
Jacques MOLNE	mon_4	4096	F-ECM	2015-07-03 00:08	5.5	1.0369	45477879701734570611058964078361695337745924097
Jacques MOLNE	mon_4	4096	F-ECM	2015-07-02 17:33	5.2	1.0369	45477879701734570611058964078361695337745924097
Jacques MOLNE	mon_4	4096	F-ECM	2015-07-02 06:30	4.7	1.0369	45477879701734570611058964078361695337745924097

This happens if you run the F-ECM without the optional Parms of listing the known factors as defined in the readme.txt

ECM2=k,b,n,c,B1,B2,curves_to_run[,"comma-separated-list-of-known-factors"]

[ewmayer]

Originally Posted by axn

F-ECM stands for Fermat-ECM. This is a result of ECM run on 2^4096+1. Incidentally, the factor found is a composite one consisting of some of the previously-known factors for that number. As for "should it be there in the cleared exponents list", I'll let the project admins make that decision.

In reponse to the thread-titular question I was gonna say "since base-25", but your answer is better. ;)

[ATH]

Originally Posted by ewmayer

In reponse to the thread-titular question I was gonna say "since base-25", but your answer is better. ;)

That is interesting, I wonder if there is always a base where any string of lets say 0-9 or 0-F (hexidecimal) is prime.

Edit: Except 0 and 1 I guess.

[axn]

Originally Posted by ATH

That is interesting, I wonder if there is always a base where any string of lets say 0-9 or 0-F (hexidecimal) is prime.

No. Note that the string of digits can be represented as a polynomial in b = d_n.b^n + ... d₀ where d_i are the digits. For the given set of digits, then, if the polynomial is reducible, then no base can yield a prime. If the polynomial is not reducible, we would expect some base (infinite number of bases?) to yield a prime.

Consider the polynomial b^4 + 8*b^2 + 15 = (b^2+3)*(b^2+5) represented by the digits 1080F. This can never yield a prime (not even in negative bases).

[Batalov]

Originally Posted by ATH

That is interesting, I wonder if there is always a base where any string of lets say 0-9 or 0-F (hexidecimal) is prime.

Edit: Except 0 and 1 I guess.

An example for "Except 0 and 1":
252_b = 21_b * 12_b and is never prime.