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Brocard's problem asks to find the values of n for which n!+1 is a square number m^2, where n! is the factorial (Brocard 1876, 1885). The only known solutions are n=4, 5, and ...

Four circles c_1, c_2, c_3, and c_4 are tangent to a fifth circle or a straight line iff T_(12)T_(34)+/-T_(13)T_(42)+/-T_(14)T_(23)=0. (1) where T_(ij) is the length of a ...

The confluent hypergeometric function of the second kind gives the second linearly independent solution to the confluent hypergeometric differential equation. It is also ...

Given relatively prime integers p and q (i.e., (p,q)=1), the Dedekind sum is defined by s(p,q)=sum_(i=1)^q((i/q))(((pi)/q)), (1) where ((x))={x-|_x_|-1/2 x not in Z; 0 x in ...

The Dirac matrices are a class of 4×4 matrices which arise in quantum electrodynamics. There are a variety of different symbols used, and Dirac matrices are also known as ...

Pick three points P=(x_1,y_1), Q=(x_2,y_2), and R=(x_3,y_3) distributed independently and uniformly in a unit disk K (i.e., in the interior of the unit circle). Then the ...

The double factorial of a positive integer n is a generalization of the usual factorial n! defined by n!!={n·(n-2)...5·3·1 n>0 odd; n·(n-2)...6·4·2 n>0 even; 1 n=-1,0. (1) ...

The Engel expansion, also called the Egyptian product, of a positive real number x is the unique increasing sequence {a_1,a_2,...} of positive integers a_i such that ...

In continuum percolation theory, the so-called germ-grain model is an obvious generalization of both the Boolean and Boolean-Poisson models which is driven by an arbitrary ...

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called ...