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A076670
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Prime divisors of (10^9)^(10^9) + 1 = 10^9000000000 + 1.
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1
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39937, 64513, 921601, 1514497, 9188353, 11059201, 23500801, 25159681, 99328001, 288000001, 302078977, 593920001, 864000001, 14400000001, 16002416641, 27769098241, 35904000001, 61120000001, 61600000001, 90708480001, 164457013249
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OFFSET
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1,1
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COMMENTS
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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
Contains 1137797098931682858642433, 3611707318387778163302401. - Max Alekseyev, Sep 10 2023
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REFERENCES
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NZ Science Monthly Bulletin Board, advert., 2000.
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LINKS
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EXAMPLE
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a(1) = 39937 because 39937 is the smallest prime divisor of (10^9)^(10^9) + 1.
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MATHEMATICA
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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}]
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PROG
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(PARI) forstep(p=1, 10^14, 1024, if(!ispseudoprime(p), next); if(Mod(10, p)^9000000000==-1, print(p)); )
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CROSSREFS
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Cf. A055386 (least prime factor of (2n)^(2n) + 1 ).
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KEYWORD
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nonn,fini
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AUTHOR
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EXTENSIONS
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a(20)-a(31) from Max Alekseyev, Apr 28 2013, Jul 02 2013, Sep 10 2023
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STATUS
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approved
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