A298210 Smallest n such that A001542(a(n)) == 0 (mod n), i.e., x=A001541(a(n)) and y=A001542(a(n)) is the fundamental solution of the Pell equation x^2 - 2*(n*y)^2 = 1.

1, 1, 2, 2, 3, 2, 3, 4, 6, 3, 6, 2, 7, 3, 6, 8, 4, 6, 10, 6, 6, 6, 11, 4, 15, 7, 18, 6, 5, 6, 15, 16, 6, 4, 3, 6, 19, 10, 14, 12, 5, 6, 22, 6, 6, 11, 23, 8, 21, 15, 4, 14, 27, 18, 6, 12, 10, 5, 10, 6, 31, 15, 6, 32, 21, 6, 34, 4, 22, 3, 35, 12, 18, 19, 30

Offset

1,3

Comments

The fundamental solution of the Pell equation x^2 - 2*(n*y)^2 = 1, is the smallest solution of x^2 - 2*y^2 = 1 satisfying y == 0 (mod n).

If n is prime (i.e., n in A000040) then a(n) divides (n - Legendre symbol (n/2)); the Legendre symbol (n/2), or more general Kronecker symbol (n/2) is A091337(n). - A.H.M. Smeets, Jan 23 2018

From A.H.M. Smeets, Jan 23 2018: (Start)

Stronger, but conjectured:

If n is prime (i.e., in A000040) and n in {2,3,5,7,11,13,19,23} (mod 24) then (n - Legendre symbol (n/2)) / a(n) == 2 (mod 4).

If n is a safe prime (i.e., in A005385) and n in {7,23} (mod 24) then (n - Legendre symbol (n/2)) / a(n) = 2, i.e., a(n) is a Sophie Germain prime (A005384).

If n is prime (i.e., in A000040) and n in {1,17} (mod 24) then (n - Legendre symbol (n/2)) / a(n) == 0 (mod 4). (End)

References

Michael J. Jacobson, Jr. and Hugh C. Williams, Solving the Pell Equation, Springer, 2009, pages 1-17.

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No. 2, Feb. 2002, pp. 182-192.

Formula

a(n) <= A000010(n) < n. - A.H.M. Smeets, Jan 23 2018

A001541(a(n)) = A002350(2*n^2).

A001542(a(n)) = A002349(2*n^2).

if n | m then a(n) | a(m).

a(2^(m+1)) = 2^m for m>=0.

Mathematica

b[n_] := b[n] = Switch[n, 0, 0, 1, 2, _, 6 b[n - 1] - b[n - 2]];
a[n_] := For[k = 1, True, k++, If[Mod[b[k], n] == 0, Return[k]]];
a /@ Range[100] (* Jean-François Alcover, Nov 16 2019 *)

Prog

(Python)
xf, yf = 3, 2
x, n = 2*xf, 0
while n < 20000:
    n = n+1
    y1, y0, i = 0, yf, 1
    while y0%n != 0:
        y1, y0, i = y0, x*y0-y1, i+1
    print(n, i)

Crossrefs

Cf. A298211, A298212.

Sequence in context: A337102 A239495 A348163 * A045772 A294977 A091256

Adjacent sequences: A298207 A298208 A298209 * A298211 A298212 A298213

Keyword

nonn

Author

A.H.M. Smeets, Jan 15 2018

Status

approved