## Primes in the form 2^n+3^n

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• What about the form 2^n+3^n ? Do primes exist in this form? There are sites about this form? Thank you Regards Giovanni Di Maria
Message 1 of 12 , Oct 22 11:23 AM
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What about the form 2^n+3^n ?
Do primes exist in this form?
Thank you
Regards
Giovanni Di Maria
• Hi Giovanni, ... 2^0 + 3^0 = 2 is a prime! 2^1 + 3^1 = 5 is a prime! 2^2 + 3^2 = 13 is a prime! ... proof that it works for all values of n is left as an
Message 2 of 12 , Oct 22 12:27 PM
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Hi Giovanni,

> What about the form 2^n+3^n ?
> Do primes exist in this form?

2^0 + 3^0 = 2 is a prime!
2^1 + 3^1 = 5 is a prime!
2^2 + 3^2 = 13 is a prime!
... proof that it works for all values of 'n' is left as an exercise for

Now, on a more serious note. The only other value of n which produces a
prime and which I know is n=4 (2^4 + 3^4 = 97). Other than that, this
form is going to be very very rare -- since if X is an odd number,
(a^X + b^X) = (a+b)(a^(X-1).b^0 - a^(X-2).b^1 + ... +/- a^0.b^(X-1)).

Thus, if 'n' has any odd factor, the result is a composite. Put slightly
differently, it'd better be a power of two if the result is to be prime.
Fermat primes would be proud of their cousins. :-)

Peter
• ... They are one form of generalised fermat primes. They have algebraic factors if n has any odd factors, and so can only be prime if n is a power of 2. (Just
Message 3 of 12 , Oct 22 12:45 PM
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> What about the form 2^n+3^n ?
> Do primes exist in this form?

They are one form of generalised fermat primes. They have algebraic factors if n has any odd factors, and so can only be prime if n is a power of 2. (Just like Fermat Numbers.)

Phil*
• Is there any place online that collects factorizations of numbers of the form a^(2^x)+b^(2^x) ? Just wondering; I just spent a few CPU minutes finding the
Message 4 of 12 , Oct 22 1:24 PM
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Is there any place online that collects factorizations of numbers of
the form a^(2^x)+b^(2^x) ?

Just wondering; I just spent a few CPU minutes finding the complete
factorization of the 123-digit number:

2^256 + 3^256 ==
72222721 *
343200070657 *
2226198380033 *
3376663028737 *
1839605176202823817996787333633 *
405549420455750246193993361998354279613273199617

and it seems like one of those things that people like to put into
online databases...

On 10/22/2010 12:45 PM, Phil Carmody wrote:
>> What about the form 2^n+3^n ?
>> Do primes exist in this form?
>
> They are one form of generalised fermat primes. They have algebraic factors if n has any odd factors, and so can only be prime if n is a power of 2. (Just like Fermat Numbers.)
>
> Phil*
>
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>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
>
>
>
• ... That was already in http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt which also reveals that 2^(2^9) + 3^(2^9) == 4043777 * 57987375533057 *
Message 5 of 12 , Oct 22 2:40 PM
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Jack Brennen <jfb@...> wrote:

> 2^256 + 3^256 ==
> 72222721 *
> 343200070657 *
> 2226198380033 *
> 3376663028737 *
> 1839605176202823817996787333633 *
> 405549420455750246193993361998354279613273199617

http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt
which also reveals that

2^(2^9) + 3^(2^9) ==
4043777 *
57987375533057 *
406297848379393 *
296909426637032277938768757025793 *
1440276254252769698681054259953238091664449537 *
2236335363090199099071989377913382509011409921 *
212085278921429005793934248630969041682071162298569605814071345111198398726196355614721

David
Message 6 of 12 , Oct 22 7:55 PM
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Jack Brennen wrote:
> Is there any place online that collects factorizations of numbers of
> the form a^(2^x)+b^(2^x) ?

http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/S0025-5718-98-00891-6.pdf
The paper is from 1998. See page 8.

--
Jens Kruse Andersen
• I had a look on the factorisation site http://www.alpertron.com.ar/ECM.HTM and it easily found the smallest factor of the 489-digit number 2^512+3^512.
Message 7 of 12 , Oct 24 6:07 PM
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I had a look on the factorisation site "http://www.alpertron.com.ar/ECM.HTM" and it easily found the smallest factor of the 489-digit number 2^512+3^512. However, so far I have not found a further factor of the 477-digit cofactor, although it certainly is not prime.

I know well that F9(10) or 10^512+1 is not yet completely factored (in fact, I once tried to spend several nights on my mediocre home computer trying to find more than the three known factors) so it would seem reasonable that 2^(2^10)+3^(2^10) would also have not been fully factored. Is any other prime factor of 2^(2^10)+3^(2^10) known?
• ... In http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt you will find the complete factorization:
Message 8 of 12 , Oct 25 1:55 AM
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> I had a look on the factorisation site "http://www.alpertron.com.ar/ECM.HTM"
> and it easily found the smallest factor of the 489-digit number 2^512+3^512.
> However, so far I have not found a further factor of the 477-digit cofactor,
> although it certainly is not prime.

In http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt you
will find the complete factorization:

4043777.57987375533057.406297848379393.296909426637032277938768757025793.1440276254252769698681054259953238091664449537.2236335363090199099071989377913382509011409921. P87

Paul
• ... The original poster seemed to write 512 where he means 2^10. It is (2^1024+3^1024)/2330249132033 that has 477 digits and is not factorized at Paul s fine
Message 9 of 12 , Oct 25 2:57 AM
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Paul Leyland <paul@...> wrote:

> > of the 477-digit cofactor
> In http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt you
> will find the complete factorization:

The original poster seemed to write 512 where he means 2^10.

It is (2^1024+3^1024)/2330249132033 that has 477 digits
and is not factorized at Paul's fine site.

David
• ... Ah. Ask and ye shall receive, but first make sure you know what you are asking for! I m running a few curves on 2^1024+3^1024 and will post again if
Message 10 of 12 , Oct 25 3:20 AM
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On Mon, 2010-10-25 at 09:57 +0000, djbroadhurst wrote:
>
> Paul Leyland <paul@...> wrote:
>
> > > so far I have not found a further factor
> > > of the 477-digit cofactor
> > In http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt you
> > will find the complete factorization:
>
> The original poster seemed to write 512 where he means 2^10.
>
> It is (2^1024+3^1024)/2330249132033 that has 477 digits
> and is not factorized at Paul's fine site.
>
> David

Ah. Ask and ye shall receive, but first make sure you know what you are

I'm running a few curves on 2^1024+3^1024 and will post again if
anything turns up.

Paul
• 2010/10/25 djbroadhurst ... http://www1.uni-hamburg.de/RRZ/W.Keller/GFNsmall.html gives one more prime factor:
Message 11 of 12 , Oct 25 3:38 AM
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>
>
>
>
> Paul Leyland <paul@...> wrote:
>
> > > so far I have not found a further factor
> > > of the 477-digit cofactor
> > In http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt you
> > will find the complete factorization:
>
> The original poster seemed to write 512 where he means 2^10.
>
> It is (2^1024+3^1024)/2330249132033 that has 477 digits
> and is not factorized at Paul's fine site.
>
> David
>
>
>
http://www1.uni-hamburg.de/RRZ/W.Keller/GFNsmall.html gives one more prime
factor: 10176954088500686156890644481 and remains c449.

[Non-text portions of this message have been removed]
• ... Found both of those now: [2010-10-25 10:16:35 GMT] n10: probable factor returned by paul@leyland.vispa.com (mesh_4:v2.0k)! Factor=2330249132033
Message 12 of 12 , Oct 25 7:13 AM
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> > The original poster seemed to write 512 where he means 2^10.
> >
> > It is (2^1024+3^1024)/2330249132033 that has 477 digits
> > and is not factorized at Paul's fine site.

> http://www1.uni-hamburg.de/RRZ/W.Keller/GFNsmall.html gives one more prime
> factor: 10176954088500686156890644481 and remains c449.

Found both of those now:

[2010-10-25 10:16:35 GMT] n10: probable factor returned by
paul@... (mesh_4:v2.0k)! Factor=2330249132033 Method=ECM
B1=250000 Sigma=90206226
[2010-10-25 10:16:35 GMT] n10: Composite factor returned by
paul@...!
Factor=160236879228127195164826503514143273848644887617419237583589880672819224787962378613168358191054320882014589878763655598739442603012525981141289393946718413503071756942365084833216516319583697404684336969516432404909365911461501691450758224189677519380661024336610343945039247522150954814862123259434023865815579858133945351151726697231584694926774191784591080286025985084327594874779310399776681428659805398074761131932929563211305463464053349254128040769205414113291414263809 Method=ECM B1=250000 Sigma=90206226

[2010-10-25 12:33:42 GMT] n10_1: probable factor returned by pcl@anubis
(anubis5)! Factor=10176954088500686156890644481 Method=ECM B1=250000
Sigma=1927001934
[2010-10-25 12:33:42 GMT] n10_1: Composite factor returned by
pcl@anubis!
Factor=15745072428810966584413689569028050952121723170588717492356803041962045543719421466634314626515681367806038954950649466918177284600127897807223229616734493963870994176961152031838265180914651227518129736337432709373581447583480385668299052789615157720829985754126147818607443036795542666041266727368967991929574179247057328372407884419914237683940826611054069893545018041498309020902084444177674786383308058081865530163424306601965434254433964505089 Method=ECM B1=250000 Sigma=1927001934

I'll leave it in the ECMserver for the moment as it is being given a
relatively small fraction of my resources.

Paul
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