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A076670
Prime divisors of (10^9)^(10^9) + 1 = 10^9000000000 + 1.
1
39937, 64513, 921601, 1514497, 9188353, 11059201, 23500801, 25159681, 99328001, 288000001, 302078977, 593920001, 864000001, 14400000001, 16002416641, 27769098241, 35904000001, 61120000001, 61600000001, 90708480001, 164457013249
OFFSET
1,1
COMMENTS
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).
Cyclotomic factorization: 10^(9*10^9) + 1 = Product_{d|9*5^9} Phi_{1024*d}(10).
Every term is congruent to 1, 2049, 3073, or 9217 modulo 10240. - Max Alekseyev, Aug 30 2023
a(32) > 10^16. - Max Alekseyev, Sep 10 2023
Contains 1137797098931682858642433, 3611707318387778163302401. - Max Alekseyev, Sep 10 2023
REFERENCES
NZ Science Monthly Bulletin Board, advert., 2000.
LINKS
S. S. Wagstaff, The Cunningham Project.
EXAMPLE
a(1) = 39937 because 39937 is the smallest prime divisor of (10^9)^(10^9) + 1.
MATHEMATICA
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}]
PROG
(PARI) forstep(p=1, 10^14, 1024, if(!ispseudoprime(p), next); if(Mod(10, p)^9000000000==-1, print(p)); )
CROSSREFS
Cf. A055386 (least prime factor of (2n)^(2n) + 1 ).
Sequence in context: A257185 A254985 A210386 * A251757 A250334 A250909
KEYWORD
nonn,fini
AUTHOR
Donald S. McDonald, Oct 25 2002
EXTENSIONS
Thanks for help from Kurt Foster and Bob Backstrom (Australia) - Donald S. McDonald
Edited and extended by Robert G. Wilson v, Nov 13 2002
Definition corrected by Sean A. Irvine, Feb 16 2010
Definition corrected by Max Alekseyev, Apr 28 2010
a(20)-a(31) from Max Alekseyev, Apr 28 2013, Jul 02 2013, Sep 10 2023
STATUS
approved