[seqfan] Re: Q about A152926

[zak seidov]

See also:

%C A007530 Except for the first term, 5, all terms == 11 (mod 30) - Zak Seidov (zakseidov at yahoo.com),Dec 04 2008



--- On Wed, 12/17/08, f.firoozbakht at sci.ui.ac.ir <f.firoozbakht at sci.ui.ac.ir> wrote:

> From: f.firoozbakht at sci.ui.ac.ir <f.firoozbakht at sci.ui.ac.ir>
> Subject: [seqfan] Re: Q about A152926
> To: seqfan at list.seqfan.eu
> Cc: mymontain at yahoo.com
> Date: Wednesday, December 17, 2008, 3:39 AM
> Dear Zak,
> 
> We can show the stronger assertion:  all terms == 21 (mod
> 30).
> 
> Proof:
> 
> 19n+2 is prime so n is odd, namely
> n = 1 (mod 2)  (I).
> 19n+2, 19n+4 are two primes greater than 3 and 19n+2 = n-1
> <>0(mod 3) ,
>     19n+4 = n-2<>0 (mod 3) so n<>1 (mod 3) and
> n<>2 (mod 3) hence,
> n=0 (mod 3)  (II).
> Also since 19n+2, 19n+4, 19n+8 & 19n+10 are four primes
> greater than 5
> and 19n+2 = 2-n (mod 5) , 19n+4 = 4-n (mod 5)
> 19n+8 = 3-n (mod 5) & 19n+10 = -n (mod 5) hence n = 1
> (mod 5) (III).
>   From (I) & (III) we conclude that n = 1 (mod 10)
> (IV).
> And from (IV) and (II) we conclude that n is of the form
> 30k+21 namely
> n = 21 (mod 30).
> 
> 
> Farideh
> 
> 
> 
> 
> 
> 
> TheQuoting zak seidov <zakseidov at yahoo.com>:
> 
> > %C A152926 All terms == 6 (mod 15)
> > - but why?
> > thx, zak
> >
> > %I A152926
> > %S A152926
> 171,3801,5781,8721,8781,17601,18231,19011,24741,28251,40431,48951,
> > %T A152926     
> >
> 49371,58821,70521,79401,79701,83391,87321,95781,96501,99501,102861,
> > %U A152926     
> >
> 109431,123171,125061,137091,177201,220311,224511,225561,229551,242451
> > %N A152926 Numbers n with property that 19n+{2,4,
> 8,10} are two     
> > subsequent twin primes.
> > %C A152926 All terms == 6 (mod 15).
> > %e A152926 19*171+{2,4}={3251,3253} and
> 19*171+{8,10}={3257,3259}     
> > are 85th and 86th twin primes.
> > %e A152926 19*3801+{2,4}={72221,72223} and     
> > 19*3801+{8,10}={72227,72229} are 935-th and 936-th
> twin primes.
> > %Y A152926 A001359 Lesser of twin primes.
> > %K A152926 nonn
> > %O A152926 1,1
> > %A A152926 Zak Seidov (zakseidov(AT)yahoo.com), Dec 15
> 2008
> >
> >> _______________________________________________
> >
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> >
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> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
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> University of Isfahan (http://www.ui.ac.ir)
> 
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