ε = ½(t + u√d)

with integers t and u, it expresses in another form

ht/u modulo p

for any prime number p > 2 that divides d. In case p > 3 it states that

where m = d/p, χ is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here

$lfloor\; xrfloor$

represents the floor function of x.

A related result is that if p is congruent to one mod four, then

$\{u\; over\; t\}h\; equiv\; B\_\{(p-1)/2\}\; mod\; p$

where B_n is the nth Bernoulli number.

There are some generalisations of these basic results, in the papers of the authors.

References

• N. C. Ankeny, E. Artin, S.Chowla, The class-number of real quadratic number fields, Annals of Math. 56 (1953), 479-492