144 Normat 57:3, 144 (2009)

Summary in English

Marius Overholt, Summer av to

kvadrat (Norwegian).

The characterization of numbers repre-

sentable as the sum of two squares in

terms of their divisors has been known

since the 17th century (Girard, Fermat)

but not proved until 18th century (Eu-

ler). A reﬁnement involving the number

R(n) of representations of n as sum of

two squares was proved by Jacobi using

theta functions. This article starts out

by reproducing a very elementary proof

due to Heath-Brown of Fermats char-

acterization of primes of type 4n + 1.

Then it turns to discussing problems

concerning asymptotic distribution, oc-

curence in prescribed intervals, inﬁnite

occurences of patterns such as n, n +

h

1

, n + h

2

. . . h

k

for ﬁxed h

i

(It is shown

that n, n + 1, n + 2 occurs inﬁnitely of-

ten, but of course n, n + 1, n + 2, n + 3

may never all be sums of two squares.)

Those are compared with the corre-

sponding statements for primes. One

may note that the asymptotic behaviour

of B(x) =

P

n=a

2

+b

2

<x

1 is more diﬃ-

cult to prove (Landau) than the prime

number theorem, but the corresponding

problem of R(x) =

P

n=a

2

+b

2

<x

R(n)

taking into account the number R(n) of

representations of n is much easier to

handle and can naturally be linked with

elementary asymptotic formulas for the

number of lattice points inside circles as

observed by Gauss. Finally one may ex-

pect that every interval [x, x + h] con-

tains a sum of two squares if h > C log x

for some suitable constant C, but so far

it has only been shown for h > Cx

1

4

.

But if we relax the condition to almost

all such intervals, there is a complete so-

lution.

O. Znamenskaya & A. Shchuplev,

Real Toric Surfaces.

In this note is proved that every real

toric smooth orientable surface is topo-

logically a torus. The presentation is

self-contained and uses no prior knowl-

edge of toric varieties.

A. Percy and D. G. Rogers, Al-

ternative route: from van Schooten to

Ptolemy.

A cyclic quadrilateral is a polygon with

four vertices, all of which lie on a cir-

cle. Such enjoy some special properties.

It is well-known that the sum of oppo-

site angles always add up to π, it is

maybe less well-known that the rect-

angle formed by the diagonals has the

same area as the sum of the rectangles

made up by opposite sides. The latter

is known as Ptolemy’s theorem. It has

many consequences, not only of trigono-

metric computations, but also of justi-

fyinhg the elegant solution of the Dutch

mathematician van Schooten (1615-60)

of constructing the minimal sum of dis-

tances from a point to the vertices of a

triangle, a problem posed as a challenge

by Fermat. The article gives a historical

survey and indicates how van Schottens

construction could serve as an inspira-

tion by ’cutting and pasting’ to suggest

and prove Ptolemy’s theorem.

Ulf Persson, Trianglar med givna om-

skrivna och inskrivna cirklar.(Swedish)

This is a second article in a series of

three, concerning 1-dimensional families

of triangles. In this case those with given

circumscribed and inscribed radii - R, r

are presented. The construction of such

families is a special case of Poncelet’s

theorem, exploiting the approach via el-

liptic curves, initiated by Jacobi.