Euler's theorem

Main variants

• Number theory. If n is a positive integer and a is an integer coprime to n, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function. This generalizes Fermat's little theorem and follows from the multiplicative structure of the units modulo n.
• Polyhedral formula. For a convex polyhedron the numbers of vertices V, edges E and faces F satisfy V − E + F = 2. This identity led to the topological notion of the Euler characteristic, χ = V − E + F, which classifies surfaces by genus when generalized.
• Homogeneous functions. If a differentiable function f(x1,...,xn) is homogeneous of degree k (f(λx)=λk f(x)), then Σ xi ∂f/∂xi = k f(x). This Euler identity appears in calculus of several variables and has applications in economics and physics.
• Rotation theorem (rigid body geometry). Any proper rotation of three-dimensional space about the origin has an invariant axis: the rotation is equivalent to turning around some line through the origin. This underlies axis–angle representations and many parametrizations of rotations.

These statements are logically independent but share Euler's hallmark of connecting simple algebraic or combinatorial counts with deeper structure. For example, the number-theoretic result is best seen as a group-theoretic fact about the multiplicative group of integers modulo n, while the polyhedral formula anticipates topological invariants that survive continuous deformation.

Historical notes and applications

Euler stated and proved many of these results in the 18th century; subsequent work expanded them into modern algebra, topology and analysis. The number-theoretic theorem is fundamental in elementary cryptography (RSA and related protocols use φ(n)). The polyhedral formula is a starting point in algebraic topology and combinatorial geometry. The homogeneous-function identity appears in thermodynamics, production theory and homogeneous potential problems. The rotation theorem is used throughout mechanics, computer graphics and robotics.

Because the label "Euler's theorem" can refer to different facts, it is important to name the context (for example, "Euler's theorem in number theory" or "Euler characteristic"), avoiding confusion with other famous Euler results such as Euler's formula e^{iθ}=cos θ + i sin θ or Euler's solution of the Königsberg bridge problem that founded graph theory.

Overall, theorems bearing Euler's name illustrate his breadth: simple, elegant formulas that link algebra, geometry and topology, and that continue to inform both theoretical work and practical applications today.