A076670 - OEIS

%I #43 Sep 12 2023 08:32:39

%S 39937,64513,921601,1514497,9188353,11059201,23500801,25159681,

%T 99328001,288000001,302078977,593920001,864000001,14400000001,

%U 16002416641,27769098241,35904000001,61120000001,61600000001,90708480001,164457013249

%N Prime divisors of (10^9)^(10^9) + 1 = 10^9000000000 + 1.

%C Numbers of the form 10^{10h}+1 can be algebraically factored into (10^{2h}+1)*L*M, L=A-B, M=A+B, h=2k-1, A=10^{4h}+5.10^{3h}+7.10^{2h}+5.10^h+1, B=10^k(10^{3h}+2.10^{2h}+2.10^h+1).

%C Cyclotomic factorization: 10^(9*10^9) + 1 = Product_{d|9*5^9} Phi_{1024*d}(10).

%C Every term is congruent to 1, 2049, 3073, or 9217 modulo 10240. - _Max Alekseyev_, Aug 30 2023

%C a(32) > 10^16. - _Max Alekseyev_, Sep 10 2023

%C Contains 1137797098931682858642433, 3611707318387778163302401. - _Max Alekseyev_, Sep 10 2023

%D NZ Science Monthly Bulletin Board, advert., 2000.

%H Max Alekseyev, <a href="/A076670/b076670.txt">Table of n, a(n) for n = 1..31</a>

%H S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/cun/">The Cunningham Project</a>.

%e a(1) = 39937 because 39937 is the smallest prime divisor of (10^9)^(10^9) + 1.

%t NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; Do[ While[ PowerMod[10, 9000000000, p] + 1 != p, p = NextPrim[p]]; Print[p]; p++, {n, 1, 19}]

%o (PARI) forstep(p=1, 10^14, 1024, if(!ispseudoprime(p), next); if(Mod(10,p)^9000000000==-1, print(p)); )

%Y Cf. A055386 (least prime factor of (2n)^(2n) + 1 ).

%K nonn,fini

%O 1,1

%A _Donald S. McDonald_, Oct 25 2002

%E Thanks for help from Kurt Foster and Bob Backstrom (Australia) - _Donald S. McDonald_

%E Edited and extended by _Robert G. Wilson v_, Nov 13 2002

%E Definition corrected by _Sean A. Irvine_, Feb 16 2010

%E Definition corrected by _Max Alekseyev_, Apr 28 2010

%E a(20)-a(31) from _Max Alekseyev_, Apr 28 2013, Jul 02 2013, Sep 10 2023