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Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique ...

If g is a continuous function g(x) in [a,b] for all x in [a,b], then g has a fixed point in [a,b]. This can be proven by supposing that g(a)>=a g(b)<=b (1) g(a)-a>=0 ...

Let X and Y be CW-complexes, and let f:X->Y be a continuous map. Then the cellular approximation theorem states that any such f is homotopic to a cellular map. In fact, if ...

Given n circles and a perimeter p, the total area of the convex hull is A_(Convex Hull)=2sqrt(3)(n-1)+p(1-1/2sqrt(3))+pi(sqrt(3)-1). Furthermore, the actual area equals this ...

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal. ...

Let n-1=FR where F is the factored part of a number F=p_1^(a_1)...p_r^(a_r), (1) where (R,F)=1, and R<sqrt(n). Pocklington's theorem, also known as the Pocklington-Lehmer ...

The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field K that ...

A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-dimensional compact smooth C^infty boundaryless ...

Let a_n>=0 and suppose sum_(n=1)^inftya_ne^(-an)∼1/a as a->0^+. Then sum_(n<=x)a_n∼x as x->infty. This theorem is a step in the proof of the prime number theorem, but has ...

If two projective pencils of curves of orders n and n^' have no common curve, the locus of the intersections of corresponding curves of the two is a curve of order n+n^' ...