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Find consecutive powers, i.e., solutions to x^p-y^q=+/-1, excluding 0 and 1. Catalan's conjecture states that the only solution is 3^2-2^3=1, so 8 and 9 (2^3 and 3^2) are the ...

There are two identities known as Catalan's identity. The first is F_n^2-F_(n+r)F_(n-r)=(-1)^(n-r)F_r^2, where F_n is a Fibonacci number. Letting r=1 gives Cassini's ...

An object created by folding a piece of paper along certain lines to form loops. The number of states possible in an n-flexagon is a Catalan number. By manipulating the ...

A formula for the generalized Catalan number _pd_(qi). The general formula is (n-q; k-1)=sum_(i=1)^k_pd_(qi)(n-pi; k-i), where (n; k) is a binomial coefficient, although ...

A parallelogram polyomino is a polyomino such that the intersection with every line perpendicular to the main diagonal is a connected segment. The number of parallelogram ...

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 ...

Catalan's triangle is the number triangle 1 ; 1 1 ; 1 2 2 ; 1 3 5 5 ; 1 4 9 14 14 ; 1 5 14 28 42 42 ; 1 6 20 48 90 132 132 (1) (OEIS A009766) with entries given by ...

An equilibrium minimal surface for a crystal or drop which has the least anisotropic surface free energy for a given volume. It is the anisotropic analog of a sphere. In the ...

The problem in calculus of variations to find the minimal surface of a boundary with specified constraints (usually having no singularities on the surface). In general, there ...

Gaussian curvature, sometimes also called total curvature (Kreyszig 1991, p. 131), is an intrinsic property of a space independent of the coordinate system used to describe ...