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Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer.
The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [1] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2]
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,
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Precisely, Let p be an odd prime and a be an integer coprime to p .
Euler's solution of the Königsberg bridge problem is considered to be the first theorem of graph theory. In addition, his recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology .
Euclid–Euler theorem (number theory) Euler's partition theorem (number theory) Euler's polyhedron theorem ; Euler's quadrilateral theorem ; Euler's rotation theorem ; Euler's theorem (differential geometry) Euler's theorem (number theory) Euler's theorem in geometry (triangle geometry) Euler's theorem on homogeneous functions (multivariate ...
Euler's theorem: = | | = In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
The theorem can be interpreted combinatorially in terms of partitions.In particular, the left hand side is a generating function for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts.