The notions of permutable and weak-permutable convergence of a series
∑n=1∞​an​ of real numbers are introduced. Classically, these
two notions are equivalent, and, by Riemann's two main theorems on the
convergence of series, a convergent series is permutably convergent if and only
if it is absolutely convergent. Working within Bishop-style constructive
mathematics, we prove that Ishihara's principle \BDN implies that every
permutably convergent series is absolutely convergent. Since there are models
of constructive mathematics in which the Riemann permutation theorem for series
holds but \BDN does not, the best we can hope for as a partial converse to our
first theorem is that the absolute convergence of series with a permutability
property classically equivalent to that of Riemann implies \BDN. We show that
this is the case when the property is weak-permutable convergence