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Smallest k > 0 such that (5^n+k)*5^n-1 and (5^n+k)*5^n+1 are a twin prime pair.
2

%I #22 Sep 03 2024 15:03:17

%S 1,17,85,47,19,71,955,815,223,269,607,1619,2737,53,883,1319,2797,4757,

%T 1585,1535,295,557,3511,269,277,5747,2125,53,13345,2195,109,1175,2995,

%U 5597,3787,1619,6577,8549,3475,5435,4807,20045,23353,9341,2857,2117,1429,16283,25333,7949

%N Smallest k > 0 such that (5^n+k)*5^n-1 and (5^n+k)*5^n+1 are a twin prime pair.

%C Conjecture: there is always one such k for n > 0.

%H Pierre CAMI, <a href="/A212487/b212487.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = A082466(5^n). - _R. J. Mathar_, Jul 20 2012

%p A212487 := proc(n)

%p local k,p ;

%p for k from 1 do

%p p := (5^n+k)*5^n-1 ;

%p if isprime(p) and isprime(p+2) then

%p return k;

%p end if;

%p end do:

%p end proc: # _R. J. Mathar_, Jul 20 2012

%o (PFGW)

%o SCRIPT

%o DIM nn,0

%o DIM kk

%o DIM jj

%o DIMS tt

%o OPENFILEOUT myfile,a(n).txt

%o OPENFILEOUT myf,b(n).txt

%o LABEL loopn

%o SET nn,nn+1

%o SET jj,0

%o IF nn>500 THEN END

%o SET kk,-1

%o LABEL loopk

%o SET kk,kk+2

%o SETS tt,%d,%d\,;nn;kk

%o PRP (5^nn+kk)*5^nn-1,tt

%o IF ISPRP THEN GOTO a

%o IF ISPRIME THEN GOTO a

%o GOTO loopk

%o LABEL a

%o SET jj,jj+1

%o PRP (5^nn+kk)*5^nn+1,tt

%o IF ISPRP THEN GOTO d

%o IF ISPRIME THEN GOTO d

%o GOTO loopk

%o LABEL d

%o WRITE myfile,tt

%o SETS tt,%d,%d\,;nn;jj

%o WRITE myf,tt

%o GOTO loopn

%Y Cf. A212488.

%K nonn

%O 1,2

%A _Pierre CAMI_, Jul 18 2012