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A220949
Least prime p such that sum_{k=0}^n (2k+1)*x^(n-k) is irreducible modulo p.
1
2, 2, 3, 2, 5, 3, 71, 23, 11, 2, 5, 2, 13, 23, 47, 47, 269, 2, 7, 19, 53, 101, 7, 53, 113, 11, 23, 2, 43, 347, 53, 283, 191, 17, 41, 2, 239, 677, 3, 281, 37, 641, 613, 41, 17, 269, 181, 137, 383, 41, 127, 2, 71, 739, 71, 353, 59, 2, 83, 2
OFFSET
1,1
COMMENTS
Conjecture: a(n) <= n^2+22 for all n>0.
We have similar conjectures with 2k+1 in the definition replaced by (2k+1)^m (m=2,3,...).
LINKS
Zhi-Wei Sun, A family of polynomials and a related conjecture on primes, a message to Number Theory List, March 30, 2013.
EXAMPLE
a(5) = 5 since f(x) = x^5+3*x^4+5*x^3+7*x^2+9*x+11 is irreducible modulo 5, but f(x)==(x+1)*(x^2+x+1)^2 (mod 2) and f(x)==(x+1)^4*(x-1) (mod 3).
Note also that a(7) = 71 = 7^2+22.
MATHEMATICA
A[n_, x_] := A[n, x] = Sum[(2k+1)*x^(n-k), {k, 0, n}]; Do[Do[If[IrreduciblePolynomialQ[A[n, x], Modulus->Prime[k]] == True, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n^2+22]}]; Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 07 2013
STATUS
approved