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cos(pi/(16)) = 1/2sqrt(2+sqrt(2+sqrt(2))) (1) cos((3pi)/(16)) = 1/2sqrt(2+sqrt(2-sqrt(2))) (2) cos((5pi)/(16)) = 1/2sqrt(2-sqrt(2-sqrt(2))) (3) cos((7pi)/(16)) = ...

cos(pi/(30)) = 1/4sqrt(7+sqrt(5)+sqrt(6(5+sqrt(5)))) (1) cos((7pi)/(30)) = 1/4sqrt(7-sqrt(5)+sqrt(6(5-sqrt(5)))) (2) cos((11pi)/(30)) = 1/4sqrt(7+sqrt(5)-sqrt(6(5+sqrt(5)))) ...

Closed forms are known for the sums of reciprocals of even-indexed Fibonacci numbers P_F^((e)) = sum_(n=1)^(infty)1/(F_(2n)) (1) = ...

1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two ...

The Diophantine equation x^2+k=y^3, which is also an elliptic curve. The general equation is still the focus of ongoing study.

A Mandelbrot set-like fractal obtained by iterating the map z_(n+1)=z_n^3+(z_0-1)z_n-z_0.

Erdős proved that there exist at least one prime of the form 4k+1 and at least one prime of the form 4k+3 between n and 2n for all n>6.

It is possible to place 7 cigarettes in such a way that each touches every other if l/d>7sqrt(3)/2 (Gardner 1959, p. 115).

Inside a ball B in R^3, {rectifiable currents S in BL area S<=c, length partialS<=c} is compact under the flat norm.

Let K subset V subset S^3 be a knot that is geometrically essential in a standard embedding of the solid torus V in the three-sphere S^3. Let K_1 subset S^3 be another knot ...