EUREKA!!!!!!!!!!!!!!!!!!!!!!!!

[Artur]

Dear Max,
I have to 10^7 digits step by steps. But for system in basis 100 I'm going  
ten times quicker as in decimal but probability of success is every time  
1/500
that I have to do 10000000*500 =500 000 000 checkings where in decimal  
these was
275 000 000 "only"
Additional parameter in optimalization is time
In 100-base system T_100=Sum[Log[x],{x,1,500000000,100)]
in 10-base T_10=Sum[Log[x],{x,1,25000000,10}]+Sum[Log[x],{x,250000000,10)]
My problem is find as n that T_n is smallest. Optimalization problem isn't  
linear.
BEST WISHES
ARTUR

Dnia 16-12-2006 o 22:18:12 Max A. <maxale at gmail.com> napisał(a):

> If I understood you correctly, you want to win the EFF prize for a
> prime with 10^7 digits by finding a number of that size such that it
> has a prime number nearby (e.g., "3 last digits varying").
>
> I see two problems with your arguments:
>
> First, it is not clear at all why only 3 last digits will be varying.
> OK, you've got some small examples supporting this claim but examples
> do not prove anything. Have you heard of the Law of small numbers?
> http://mathworld.wolfram.com/StrongLawofSmallNumbers.html
>
> Second, even if you are correct (or lucky) to catch a prime number
> with 10^7 digits, how you are going to prove that it is prime? What is
> "modified Wilson-Lehmer" you are referring to? Generic deterministic
> primality tests are not fast enough for the numbers of that size.
> There are fast primality tests but they are probabilistic, meaning
> that they cannot substitute a proof of primality (required by EFF).
>
> Max
>
>
> On 12/16/06, Artur <grafix at csl.pl> wrote:
>> Dear Seqfans,
>> Is following optimalization problem
>> Finding basis of counting system such that time reaching 10 000 000  
>> digits
>> prime by my procedure will be shortest.
>> Formula will be follwing for my previous sequence will be:
>> If we take counting system basis 100 3 last digits varying (see bellow)
>> For my previous system if we go from a(odd) to a(even) one digit  
>> varying.
>> Probababilty of success (that we find prime) is 1/5 if we go from  
>> a(even)
>> to a(odd) two last digits varying and in this case probabilty is 1/50.  
>> To
>> reaching prime 10 000 000 digits we need 5 000 000 checkings with
>> probability 1/5 and 5 000 000 with probability 1/50 together
>> 5000000*5+5000000*50 = 275000000 steps. Now we take to counting that  
>> time
>> of checking that number is prime or not by modified Wilson-Lehmer
>> algorithm is log(number of digits). My question is which counting system
>> will be the quickest one.
>> BEST WISHES
>> ARTUR
>>
>>
>>
>>
>> %I A000001
>> %S A000001 2, 199, 19891, 1989077, 198907679, 19890767893,  
>> 1989076789283,
>> 198907678928279, 19890767892827873, 1989076789282787297,
>> 198907678928278729697, 19890767892827872969661,  
>> 1989076789282787296966091,
>> 198907678928278729696609039, 19890767892827872969660903813,
>> 1989076789282787296966090381249, 198907678928278729696609038124829,
>> 19890767892827872969660903812482509,  
>> 1989076789282787296966090381248250849,
>> 198907678928278729696609038124825084813,
>> 19890767892827872969660903812482508481211,
>> 1989076789282787296966090381248250848120823,
>> 198907678928278729696609038124825084812082207,
>> 19890767892827872969660903812482508481208220559,
>> 1989076789282787296966090381248250848120822055831,
>> 198907678928278729696609038124825084812082205583091,
>> 19890767892827872969660903812482508481208220558309043,
>> 1989076789282787296966090381248250848120822055830904249,
>> 198907678928278729696609038124825084812082205583090424757,
>> 19890767892827872969660903812482508481208220558309042475663
>> %N A000001 Biggest primes <100*a[n-1] a[1]=2
>> %F A000001 a(n) = 10*a(n-1)
>> %Y A000001 A006902,A040016,A120031-A120041
>> %O A000001 1
>> %K A000001 ,nonn,
>> %A A000001 Artur Jasinski (grafix at csl.pl), Dec 16 2006
>> Dnia 16-12-2006 o 20:16:13 Artur <grafix at csl.pl> napisał(a):
>>
>> > Dear Jonathan and Others,
>> > Binary system isn't as effective as decimal because 5 or 6 last digits
>> > varying
>> > %I A000001
>> > %S A000001
>> > 10,11,101,111,1101,10111,10101,1010011,10100011,100111101,1001110111,
>> > 10011101011,100111000111,1001110001011,10011011110101,100110111100001,
>> >  
>> 1001101110111101,10011011101010011,100110111010011101,1001101110100011111,
>> > 10011011101000111011,100110111010001101101,1001101110100010111111,
>> >  
>> 10011011101000101110101,100110111010001011010111,1001101110100010110101011,
>> > 10011011101000101101000001,100110111010001011001110011,
>> > 100110111010001011011100001,10011011101000101100110111001,
>> > 100110111010001011001101010111,1001101110100010110011010010111,
>> > 10011011101000101100110100011001,100110111010001011001101000101011,
>> >  
>> 1001101110100010110011010001000001,10011011101000101100110100001111001,
>> >  
>> 100110111010001011001101000011101111,1001101110100010110011010000110111101,
>> >  
>> 10011011101000101100110100001101011011,100110111010001011001101000011010110101
>> > %N A000001 Biggest primes writen in binary system <2*a[n-1] a[1]=2
>> > %C A000001 These sequence visualized Chebyshev low that between n and  
>> 2n
>> > existed minimum 1 prime
>> > %F A000001 a(n) = 2*a(n-1)
>> > %Y A000001 A006902,A040016,A120031-A120041
>> > %O A000001 1
>> > %K A000001 ,nonn,
>> > %A A000001 Artur Jasinski (grafix at csl.pl), Dec 16 2006
>> >
>> > Dnia 16-12-2006 o 18:12:23 Jonathan Post <jvospost3 at gmail.com>
>> > napisał(a):
>> >
>> >> "Chebychev said it and I'll say it again:
>> >> There's always a prime between n and 2n."
>> >>
>> >> I skipped the attribution in A118909.*
>> >> *
>> >> Joseph Louis François Bertrand [1822-1900] was the Paris professor  
>> who
>> >> made
>> >> the conjecture, proved by Chenychev in 1850. I have heard that It
>> >> appears
>> >> that Nat Fine wrote this couplet in honor of Paul Erdos.
>> >>
>> >> Jasinski said it, and emailed again:
>> >> There's always a prime between n and 10n.
>> >>
>> >> "ten" rhymes with "n."  This is as great an advance in mathematical
>> >> poetry
>> >> as any sequence containing "14" is to sonnets.  Or, more to the  
>> point,
>> >> as
>> >> the fact that the number of syllables in a haiku is prime.
>> >>
>> >> I'm nearly the last person to criticize anyone who submits a base
>> >> sequence,
>> >> or a sequence involving primes. But when I do, I know it to be a base
>> >> sequence, and endeavor to submit the more generalized sequence of
>> >> antidiagonals of the array of such a sequence over all natural number
>> >> bases.
>> >>
>> >> Sometimes generalization sheds new light on a problem. Grothendieck,  
>> for
>> >> instance, as a master of that, before he turned his mind to  
>> politics. If
>> >> there is an enlightening generalization that moves beyond the
>> >> arbitrariness
>> >> on decimal base, I'd be pleased to see it.
>> >>
>> >> No disrespect to Jasinski.  I also have made the mistake of
>> >> egotistically
>> >> proposing to name something after myself (a la Donald Trump), and  
>> over
>> >> premature exclamation marks, not symbolizing factorials, out of
>> >> excitement
>> >> and enthusiam. Also, sometimes I like the sequences by this  
>> gentleman,
>> >> as
>> >> indicated by A113914 and its ilk.
>> >>
>> >> In this holiday season, perhaps it is best to lean towards tolerance,
>> >> charity, forgiveness, and kindness.
>> >>
>> >> -- Jonathan Vos Post
>> >>
>> >> On 12/16/06, Hans Havermann <pxp at rogers.com> wrote:
>> >>>
>> >>> Antti Karttunen asked:
>> >>>
>> >>> So, please tell us, what is the ground-breaking idea in your primes
>> >>> below?
>> >>>
>> >>>
>> >>> 17989, 179849, 1798487, 17984833, 179848309, 1798483067,
>> >>> 17984830667, 179848306667, 1798483066669, 17984830666651,
>> >>> 179848306666507,
>> >>> ...
>> >>>
>> >>>
>> >>> I can at least verify that:
>> >>>
>> >>> a(1) = 17989
>> >>> a(n) = PreviousPrime[10*a(n-1)]
>> >>>
>> >>> I hope there's more to it than that. ;)
>> >>>
>> >>>
>> >
>> >
>> >
>> > __________ NOD32 Informacje 1924 (20061215) __________
>> >
>> > Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
>> > http://www.nod32.com lub http://www.nod32.pl
>> >
>>
>>
>>