A076670 Prime divisors of (10^9)^(10^9) + 1 = 10^9000000000 + 1.

39937, 64513, 921601, 1514497, 9188353, 11059201, 23500801, 25159681, 99328001, 288000001, 302078977, 593920001, 864000001, 14400000001, 16002416641, 27769098241, 35904000001, 61120000001, 61600000001, 90708480001, 164457013249

Offset: 1

Author

Donald S. McDonald, Oct 25 2002

Keywords

• nonn
• fini

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

Examples

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

References

• NZ Science Monthly Bulletin Board, advert., 2000.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..31
• S. S. Wagstaff, The Cunningham Project.

Crossrefs

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

Programs

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}]

PARI

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

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