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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.

%I A387207 #6 Sep 01 2025 18:27:43
%S A387207 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,
%T A387207 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,
%U A387207 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
%N 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.
%C A387207 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.
%C A387207 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.
%H A387207 Andreas Dress and Rudolf Scharlau, <a href="https://doi.org/10.1016/0022-314X(82)90064-6">Indecomposable totally positive numbers in real quadratic orders</a>, J. Number Theory 14 (1982), no. 3, 292-306.
%H A387207 Se Wook Jang and Byeong Moon Kim, <a href="https://doi.org/10.1016/j.jnt.2015.06.003">A refinement of the Dress-Scharlau theorem</a>, J. Number Theory 158 (2016), 234-243.
%H A387207 Vítězslav Kala, <a href="https://doi.org/10.1016/j.jnt.2016.02.022">Norms of indecomposable integers in real quadratic fields</a>, J. Number Theory 166 (2016), 193-207.
%H A387207 Vítězslav Kala, <a href="https://doi.org/10.46298/cm.10896">Universal quadratic forms and indecomposables in number fields: a survey</a>, Commun. Math. 31 (2023), no. 2, 81-114.
%H A387207 Magdaléna Tinková and Paul Voutier, <a href="https://doi.org/10.1016/j.jnt.2019.11.005">Indecomposable integers in real quadratic fields</a>, J. Number Theory 212 (2020), 458-482.
%F A387207 a(n) <= A005117(n) for all n >= 2 [Dress-Scharlau].
%e A387207 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.
%e A387207 For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) has norm 1, thus a(3) = 1.
%e A387207 For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) has norm 1, thus a(4) = 1.
%e A387207 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.
%o A387207 (SageMath)
%o A387207 def a(n):
%o A387207     D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
%o A387207     K.<a> = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
%o A387207     if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
%o A387207     else: cf, d = continued_fraction(a), a
%o A387207     s = len(cf.period())
%o A387207     ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
%o A387207     ind = [ai[i]+t*ai[i+1] for i in range(0, 2*s+1, 2) for t in range(cf[i+1])]
%o A387207     return max([c.norm() for c in ind])
%Y A387207 Cf. A005117, A035015, A387203.
%K A387207 nonn,new
%O A387207 2,1
%A A387207 _Robin Visser_, Aug 21 2025