Gruber P. Convex and Discrete Geometry

22 Minkowski’s First Fundamental Theorem 373
If pi = 2 put ai = 1, bi = 0. If pi is odd, each of the sets {a 2 : 0 ≤ a < pi/2}
and {−1 − b 2 : 0 ≤ b < pi/2} consists of 1 2 (pi + 1) integers which are pairwise
incongruent modulo pi. Since there are only pi residue classes modulo pi, these sets
must contain elements a 2 i and −1 − b2 i , say, which are congruent modulo pi. Then
a 2 i + b2 i + 1 ≡ 0modpi, concluding the proof of (1).
As a consequence of (1), we shall prove that:
(2) There are integers a, b such that a 2 + b 2 + 1 ≡ 0modu.
For each i, choose an integer Pi such that Pi pi = u. Since Pi and pi are relatively
prime, there is an integer qi such that Piqi ≡ 1modpi, see Proposition 21.1. Now let
Then
a = P1q1a1 +···+ Pnqnan, b = P1q1b1 +···+ Pnqnbn.
a2 + b2 + 1 ≡ P2 1 q2 1a2 1 +···+ P2 n q2 na2 n + P2 1 q2 1b2 1 +···+ P2 n q2 nb2 n + 1
≡ P 2
i q2 i (a2 i + b2 i ) + 1 ≡ a2 i + b2 i
+ 1 ≡ 0modpi,
by (1) and our choice of Pi, qi. Thus pi|a 2 + b 2 + 1 for each i. This implies that
u = p1 ···pn|a 2 + b 2 + 1, concluding the proof of (2).
Next consider the lattice L in E 4 with basis
Clearly d(L) = u 2 .Let
Then
(1, 0, a, −b), (0, 1, b, a), (0, 0, u, 0), (0, 0, 0, u).
ϱ = 2 5 4 u 1 2
π 1 2
V (ϱB 4 ) = ϱ 4 V (B 4 ) = 25 u 2
π 2
.
π 4 2
Ɣ(1 + 4 2 ) = 24 u 2 ,
where B 4 is the solid Euclidean unit ball in E 4 . An application of the fundamental
theorem, with C = ϱB 4 and the lattice L just defined, yields a point �= o of L in
ϱB 4 . Thus, there are integers v1,...,v4, not all 0, such that
0