A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

2, 1, 1, 3, 2, 10, 5, 3, 2, 1, 4, 9, 1, 11, 2, 26, 7, 6, 10, 4, 2, 1, 9, 19, 13, 10, 7, 21, 9, 2, 25, 13, 9, 7, 58, 29, 15, 2, 16, 33, 33, 3, 14, 10, 18, 74, 1, 3, 2, 82, 41, 21, 43, 13, 22, 30, 7, 18, 5, 24, 25, 51, 34, 4, 106, 53, 27, 11, 37, 28, 57, 9, 59, 2, 122, 61, 42, 16, 130, 65, 11

Offset

2,1

Comments

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

References

Oskar Perron, Die Lehre von den Kettenbruchen, Chelsea Publishing Co., New York, 1950, pp. xii+524.

Links

Robin Visser, Table of n, a(n) for n = 2..10000

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.

Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.

Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.

Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.

Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Formula

a(n) <= D if D = 2 or 3 (mod 4), else a(n) <= (D-1)/4 if D = 1 (mod 4), where D = A005117(n). Equality holds if D is a member of the sequence A003654 [Dress-Scharlau].

a(n) <= D/k if D = 2 or 3 (mod 4), else a(n) <= (D-1)/(4*k) if D = 1 (mod 4), where D = A005117(n) and k = A387618(n). Equality holds if D is a multiple of k [Jang-Kim, Theorem 5].

a(n) >= D/k - k/4 if D = 2 or 3 (mod 4), else a(n) >= (D-1)/(4*k) - k/4 if D = 1 (mod 4), where D = A005117(n) and k = A387618(n) [Jang-Kim, Corollary 3].

Example

For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) has norm either 1 or 2, thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) has norm 1, thus a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) has norm 1, thus a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) has norm either 1 or 3, thus a(5) = 3.

Prog

(SageMath)
def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K.<a> = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0, 2*s+1, 2) for t in range(cf[i+1])]
    return max([c.norm() for c in ind])

Crossrefs

Cf. A003654, A005117, A035015, A121339, A387203, A387575, A387618, A389888.

Sequence in context: A122044 A120744 A053423 * A216201 A127514 A078802

Adjacent sequences: A387204 A387205 A387206 * A387208 A387209 A387210

Keyword

nonn

Author

Robin Visser, Aug 21 2025

Status

approved