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For F_n the nth Fibonacci number, F_(n-1)F_(n+1)-F_n^2=(-1)^n. This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Wells ...

Inscribe a triangle in a circle such that the sides of the triangle pass through three given points A, B, and C.

"Casting out nines" is an elementary check of a multiplication which makes use of the congruence 10^n=1 (mod 9). Let decimal numbers be written a=a_n...a_2a_1a_0, ...

A method for verifying the correctness of an arithmetical operation on natural numbers, based on the same principle as casting out nines. The methods of sevens takes ...

If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

A catacaustic is a curve that is the envelope of rays emanating from a specified point (or a point at infinite distance producing parallel rays) for a given mirror shape. The ...

Catalan (1876, 1891) noted that the sequence of Mersenne numbers 2^2-1=3, 2^3-1=7, and 2^7-1=127, and (OEIS A007013) were all prime (Dickson 2005, p. 22). Therefore, the ...

Special cases of general formulas due to Bessel. J_0(sqrt(z^2-y^2))=1/piint_0^pie^(ycostheta)cos(zsintheta)dtheta, where J_0(z) is a Bessel function of the first kind. Now, ...

The Catalan numbers on nonnegative integers n are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular n-gon be divided into ...

The dual polyhedra of the Archimedean solids, given in the following table. They are known as Catalan solids in honor of the Belgian mathematician who first published them in ...