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This is sometimes knows as the "bars and stars" method. Suppose a recipe called for 5 pinches of spice, out of 9 spices. Each possibility is an arrangement of 5 spices ...

Count the number of lattice points N(r) inside the boundary of a circle of radius r with center at the origin. The exact solution is given by the sum N(r) = ...

In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number h(-d) of an imaginary quadratic field with binary quadratic form discriminant ...

For a given m, determine a complete list of fundamental binary quadratic form discriminants -d such that the class number is given by h(-d)=m. Heegner (1952) gave a solution ...

The reciprocal of the arithmetic-geometric mean of 1 and sqrt(2), G = 2/piint_0^11/(sqrt(1-x^4))dx (1) = 2/piint_0^(pi/2)(dtheta)/(sqrt(1+sin^2theta)) (2) = L/pi (3) = ...

Gauss's continued fraction is given by the continued fraction ...

Let p be an odd prime and b a positive integer not divisible by p. Then for each positive odd integer 2k-1<p, let r_k be r_k=(2k-1)b (mod p) with 0<r_k<p, and let t be the ...

Let p>3 be a prime number, then 4(x^p-y^p)/(x-y)=R^2(x,y)-(-1)^((p-1)/2)pS^2(x,y), where R(x,y) and S(x,y) are homogeneous polynomials in x and y with integer coefficients. ...

At rational arguments p/q, the digamma function psi_0(p/q) is given by psi_0(p/q)=-gamma-ln(2q)-1/2picot(p/qpi) +2sum_(k=1)^([q/2]-1)cos((2pipk)/q)ln[sin((pik)/q)] (1) for ...

If a sequence of double points is passed as a closed curve is traversed, each double point appears once in an even place and once in an odd place.