A100020 a(n) = smallest prime p such that x^2-n has roots in the p-adic integers.

2, 7, 11, 2, 11, 5, 3, 7, 2, 3, 5, 11, 3, 5, 7, 2, 2, 7, 3, 11, 5, 3, 7, 5, 2, 5, 11, 3, 5, 7, 3, 7, 2, 3, 13, 2, 3, 11, 5, 3, 2, 11, 3, 5, 11, 3, 11, 11, 2, 7, 5, 3, 7, 5, 3, 5, 2, 3, 5, 7, 3, 13, 3, 2, 2, 5, 3, 2, 5, 3, 5, 7, 2, 5, 11

Offset

1,1

Links

Table of n, a(n) for n=1..75.

Example

a(6)=5 because x^2-6 has roots in the 5-adic integers. Roots are
4+5+4*5^2+2*5^4+3*5^5+2*5^6+5^7+3*5^8+O(5^9) and
1+3*5+4*5^3+2*5^4+5^5+2*5^6+3*5^7+5^8+O(5^9); but this is irreducible over Qp for p in {2,3} (x^2-6 is Eisenstein for p=2 and 3).

Maple

A100020 := proc(n)
    local p, anz ;
    p := 1 ;
    anz := 0 ;
    while anz =0 do
        p := nextprime(p) ;
        poly := x^2-n ;
        anz := nops([padic[rootp](poly, p)]);
    end do:
    p ;
end proc:
seq(A100020(n), n=1..100) ;

Crossrefs

Cf. A099408.

Sequence in context: A110739 A179117 A133154 * A241807 A020638 A091385

Adjacent sequences: A100017 A100018 A100019 * A100021 A100022 A100023

Keyword

nonn

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 19 2004

Status

approved