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In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. This condition turns out also to be sufficient—a result stated by Euler and later proved by Carl Hierholzer. Such a walk is now called an Eulerian trail or Euler walk in his honor ...
The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. [1] This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian ...
Euler's identity therefore states that the limit, as n approaches infinity, of (+) is equal to −1. This limit is illustrated in the animation to the right. Euler's formula for a general angle. Euler's identity is a special case of Euler's formula, which states that for any real number x,
This proof, due to Euler, [3] uses induction to prove the theorem for all integers a ≥ 0. The base step, that 0 p ≡ 0 (mod p ) , is trivial. Next, we must show that if the theorem is true for a = k , then it is also true for a = k + 1 .
Thus, it is often called Euler's phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, [14] [15] so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's. The cototient of n is defined as n − φ(n).
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The pentagonal number theorem occurs as a special case of the Jacobi triple product. Q-series generalize Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms.