login
Integers k such that 10^k+21 is prime.
28

%I #42 Aug 30 2021 03:10:28

%S 1,3,9,17,55,77,133,195,357,1537,2629,3409,8007,25671,48003,55811,

%T 94983

%N Integers k such that 10^k+21 is prime.

%C There cannot be any primes of this form when k is even, because all such numbers must be divisible by 11. A number is divisible by 11 if the difference between the sum of its odd digits and the sum of its even digits is 0 or divisible by 11. When k is even, the difference is always 0. - _Dmitry Kamenetsky_, Jul 12 2008

%C The next term, if one exists, is >100000. - _Robert Price_, Mar 24 2011

%C See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "10021".

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/prime/">List of near-repdigit-related prime numbers</a>

%H <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>

%e For k=3 we have 10^3+21 = 1000+21 = 1021, which is prime.

%t q=21; s=""; For[ a=q,a<=q,s="10^n+"<>ToString[ a ]<>":"; n=0; For[ i=1,i< 10^3,If[ PrimeQ[ 10^i+a ],n=1; s=s<>ToString[ i ]<>"," ]; i++ ]; If[ n>0,Print[ s ] ]; a++ ] (* _Vladimir Joseph Stephan Orlovsky_, May 06 2008 *)

%o (PARI) for(n=1,1e4,if(ispseudoprime(10^n+21),print1(n", "))) \\ _Charles R Greathouse IV_, Jul 20 2011

%Y Cf. A049054, A088274, A088275, A138861.

%K more,nonn

%O 1,2

%A Julien Peter Benney (jpbenney(AT)ftml.net), Jun 01 2005

%E a(6)=77 inserted by _Vladimir Joseph Stephan Orlovsky_, May 06 2008

%E a(13)=8007 from _Dmitry Kamenetsky_, Jul 12 2008

%E a(14)=25671 from _Robert Price_, Nov 08 2010

%E Edited by _Ray Chandler_, Dec 24 2010

%E a(15)=48003 from _Robert Price_, Dec 31 2010

%E a(16)=55811 from _Robert Price_, Jan 09 2011

%E a(17)=94983 from _Robert Price_, Mar 24 2011