Topology of Numbers · Problem 4

Representations by Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · representations-by-quadratic-forms

Problem

This problem will show how things can be more complicated than in the previous problems. (a) Show that the number of equivalence classes of forms of discriminant

-23

is 2 while the number of proper equivalence classes is 3, and find reduced forms

Q_1

and

Q_2

of discriminant

-23

that are not equivalent. (b) Draw the topographs of

Q_1

and

Q_2

up to the value 70. (Again you do not have to repeat symmetric parts.) (c) Find a number

n

that occurs in both topographs, and find the

x

and

y

values that give

Q_1(x_1, y_1) = n = Q_2(x_2, y_2)

. (This sort of thing never happens in the previous problems.) (d) Find a prime

p_1

in the topograph of

Q_1

and a different prime

p_2

in the topograph of

Q_2

such that

p_1

and

p_2

are congruent mod

|\Delta| = 23

. (This sort of thing also never happens in the previous problems.)

Related theory

Binary Quadratic Forms Introduction

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