Example
The Hermite constant is known in dimensions 1–8 and 24.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
|---|---|---|---|---|---|---|---|---|---|
For n = 2, one has γ2 = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers. [1]
In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γn is the maximum of λ1(L) over all such lattices L.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
The Hermite constant is known in dimensions 1–8 and 24.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
|---|---|---|---|---|---|---|---|---|---|
For n = 2, one has γ2 = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers. [1]
It is known that [2]
A stronger estimate due to Hans Frederick Blichfeldt [3] is [4]
where is the gamma function.
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
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