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Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well. [1]: p. 106 Exactly one edge and two face diagonals of a primitive Euler brick are odd. At least two edges of an Euler brick are divisible by 3.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Euler number (algebraic topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron. Euler number (3-manifold topology) – see Seifert fiber space; Lucky numbers of Euler [4] Euler's constant gamma (γ), also known as the Euler–Mascheroni constant
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In the 1950s, Evgenii Landis and Ivan Petrovsky published a purported solution, but it was shown wrong in the early 1960s. [9] In 1954 Zarankiewicz claimed to have solved Turán's brick factory problem about the crossing number of complete bipartite graphs, but Kainen and Ringel later noticed a gap in his proof. Complex structures on the 6-sphere.
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 ...
In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. [91] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each ...
Landau's problems; Lander, Parkin, and Selfridge conjecture; Legendre's conjecture; Lehmer's conjecture; Lehmer's totient problem; Lemoine's conjecture; Leopoldt's conjecture; Lindelöf hypothesis; Lonely runner conjecture; Lychrel number