[seqfan] Re: Q about A152926

[f.firoozbakht at sci.ui.ac.ir]

The proof is similar.

Quoting zak seidov <zakseidov at yahoo.com>:

> 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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