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Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.
+10
25
3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2
COMMENTS
Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014 Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014
FORMULA
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
EXAMPLE
a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.
MAPLE
f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n, k)) then return k fi od
end proc:
MATHEMATICA
Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)
PROG
(PARI) a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014
CROSSREFS
Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980. Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.
a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime.
+10
4
2, 2, 2, 5, 4, 2, 5, 196, 14, 129, 424, 484, 22, 5164, 7726, 13325, 96873, 192098, 712012, 123447
COMMENTS
Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n. Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
The numbers of this form are Generalized Unique primes (see Links section).
a(16)=96873 corresponds to a prime with 653552 decimal digits.
The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. - Serge Batalov, Jan 15 2018 a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. - Serge Batalov, Jan 15 2018 a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. - Serge Batalov, May 31 2023
MATHEMATICA
Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n) - 1) + 1] &], {n, 0, 9}] (* Michael De Vlieger, Jan 15 2018 *)
PROG
(PARI)
a(n)=k=1; while(!ispseudoprime(k^(2^n)*(k^(2^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++) \\ Derek Orr, Aug 14 2014
Least k such that k^(3^n)*(k^(3^n) - 1) + 1 is prime.
+10
4
2, 6, 7, 93, 15, 372, 421, 759, 7426, 9087
COMMENTS
Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=3^n, which is similar to A153438. Search limits: a(10) > 35000, a(11) > 3500.
EXAMPLE
When k = 7, k^18 - k^9 + 1 is prime. Since this isn't prime for k < 7, a(2) = 7.
MATHEMATICA
a246120[n_Integer] := Module[{k = 1},
While[! PrimeQ[k^(3^n)*(k^(3^n) - 1) + 1], k++]; k]; a246120 /@ Range[0, 9] (* Michael De Vlieger, Aug 15 2014 *)
PROG
(PARI)
a(n)=k=1; while(!ispseudoprime(k^(3^n)*(k^(3^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++) \\ Derek Orr, Aug 14 2014
Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.
+10
3
2, 3, 88, 28, 688, 7003, 1925
COMMENTS
Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=6^n. Numbers of this form are Generalized unique primes. a(6) generates a 306477-digit prime.
EXAMPLE
When k = 88, k^72 - k^36 + 1 is prime. Since this isn't prime for k < 88, a(2) = 88.
PROG
(PARI) a(n)=k=1; while(!ispseudoprime(k^(6^n)*(k^(6^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++)
Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).
+10
1
2, 4, 2, 6, 2, 20, 20, 26, 25, 10, 14, 5, 373, 4, 65, 232, 56, 2, 521, 911, 1156, 1619, 647, 511, 34, 2336, 2123, 1274, 2866, 951, 2199, 1353, 4965, 7396, 13513, 3692, 14103, 32275, 2257, 86, 3928, 2779, 18781, 85835, 820, 16647, 2468, 26677, 1172, 38361, 40842
COMMENTS
1 - m^k + m^(2*k) - m^(3^k) + m^(4*k) equals Phi(10*k,m).
First 15 terms were generated by the provided Mathematica program. All other terms found using OpenPFGW as Fermat and Lucas PRP. Term 16-20, 22-24, 27 have N^2-1 factored over 33.3% and proved using OpenPFGW;
terms 21, 25, 29-33, 36, 37, 39, 41, 42, 45, 48, 51 are proved using CHG pari script;
terms 26, 28, 34, 40 are proved using kppm PARI script;
terms 35, 38, 43, 44, 46, 47, 49, 50 do not yet have a primality certificate.
The corresponding prime number of term 51 (40842) has 236089 digits.
The corresponding prime numbers for the following terms are equal:
p(3) = p(2) = Phi(10, 2^4),
p(12) = p(9) = Phi(10, 5^50),
p(18) = p(14) = Phi(10, 2^160),
p(25) = p(21) = Phi(10, 34^512),
p(40) = p(34) = Phi(10, 86^4000).
EXAMPLE
n=1, A003592(1) = 1, when a=2, 1 - 2^1 + 2^2 - 2^3 + 2^4 = 11 is prime, so a(1)=2; n=2, A003592(2) = 2, when a=4, 1 - 4^2 + 4^4 - 4^6 + 4^8 = 61681 is prime, so a(2)=4; ...
n=13, A003592(13) = 64, when a=373, PrimeQ(1 - 373^64 + 373^128 - 373^192 + 373^256) = True, while for a = 2..372, PrimeQ(1 - a^64 + a^128 - a^192 + a^256) = False, so a(13)=373.
MATHEMATICA
fQ[n_] := PowerMod[10, n, n] == 0; a = Select[10 Range@100, fQ]/10; l = Length[a]; Table[m = a[[j]]; i = 1; While[i++; cp = 1 - i^m + i^(2*m)-i^(3*m)+i^(4*m); ! PrimeQ[cp]]; i, {j, 1, l}]
PROG
(PARI) do(k)=my(m=1); while(!ispseudoprime(polcyclo(10*k, m++)), ); m
list(lim)=my(v=List(), N); for(n=0, log(lim)\log(5), N=5^n; while(N<=lim, listput(v, N); N<<=1)); apply(do, vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 04 2012
EXTENSIONS
Term 50 added and comments updated by Lei Zhou, Jul 27 2012 Term 51 added and comments updated by Lei Zhou, Oct 10 2012
a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.
+10
1
6, 3, 3, 6, 5, 106, 207, 569, 224, 736, 2854, 21234, 14837, 165394, 24743, 62721, 237804, 143332
COMMENTS
a(13) = 165394 is a significant outlier from the generally expected trend, which can be conjectured to be 6*2^n*gamma, where gamma is the Euler-Mascheroni constant A001620. Additionally, the next b > a(13) such that b^(6*2^n) - b^(3*2^n) + 1 is prime is 165836, which is remarkably close to a(13). - Serge Batalov, Jan 24 2018
EXAMPLE
2^12 - 2^6 + 1 = 4033 is composite and 3^12 - 3^6 + 1 = 530713 is prime, so a(1) = 3.
PROG
(PARI) for(n=0, 9, for(b=2, 1000, x=b^(3*2^n); if(isprime(x*(x-1)+1), print1(b, ", "); break)))
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