

A076670


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


0



39937, 64513, 921601, 1514497, 9188353, 11059201, 23500801, 25159681, 99328001, 288000001, 302078977, 593920001, 864000001, 14400000001, 16002416641, 27769098241, 35904000001, 61120000001, 61600000001, 90708480001, 164457013249, 249832960001, 16281309920257, 16598085949441, 22574752000001, 39315840000001, 132379043573761, 182544000000001, 230846400000001
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OFFSET

1,1


COMMENTS

Numbers of the form 10^{10h}+1 can be algebraically factored into (10^{2h}+1)*L*M, L=AB, M=A+B, h=2k1, 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 = \prod_{d9*5^9} \Phi_{1024*d}(10).
Every term is congruent to 1 modulo 1024.  Max Alekseyev, Apr 28 2013
a(30) > 10^15.  Max Alekseyev, Jul 02 2013
Contains 3611707318387778163302401 (a factor of 10^512+1).  Max Alekseyev, Jan 07 2015


REFERENCES

NZ Science Monthly Bulletin Board, advert., 2000.


LINKS

Table of n, a(n) for n=1..29.
S. S. Wagstaff, The Cunningham Project


EXAMPLE

a(1)= 39937 because 39937 divides (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
Adjacent sequences: A076667 A076668 A076669 * A076671 A076672 A076673


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(26) from Max Alekseyev, Apr 28 2013
a(27)a(29) from Max Alekseyev, Jul 02 2013


STATUS

approved



