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The variety which is an invariant of degree four and is given by the equation y_0^4-y_0(y_1^3+y_2^3+y_3^3+y_4^3)+3y_1y_2y_3y_4=0.

The only known classically known algebraic curve of curve genus g>1 that has an explicit parametrization (x(t),y(t)) in terms of standard special functions (Burnside 1893, ...

The quartic surface obtained by replacing the constant b in the equation of the Cassini ovals with b=z, obtaining [(x-a)^2+y^2][(x+a)^2+y^2]=z^4. (1) As can be seen by ...

Let r and s be positive integers which are relatively prime and let a and b be any two integers. Then there is an integer N such that N=a (mod r) (1) and N=b (mod s). (2) ...

The circumsphere of given set of points, commonly the vertices of a solid, is a sphere that passes through all the points. A circumsphere does not always exist, but when it ...

The conchoid of de Sluze is the cubic curve first constructed by René de Sluze in 1662. It is given by the implicit equation (x-1)(x^2+y^2)=ax^2, (1) or the polar equation ...

An equation of the form y=ax^3+bx^2+cx+d, (1) where the three roots of the equation coincide (and are therefore real), i.e., y=a(x-r)^3=a(x^3-3rx^2-3r^2x-r^3). (2) Loomis ...

A hexagon (not necessarily regular) on whose polygon vertices a circle may be circumscribed. Let sigma_i=Pi_i(a_1^2,a_2^2,a_3^2,a_4^2,a_5^2,a_6^2) (1) denote the ith-order ...

The inversion of a convolution equation, i.e., the solution for f of an equation of the form f*g=h+epsilon, given g and h, where epsilon is the noise and * denotes the ...

A pair of prime numbers (p,q) such that p^(q-1)=1 (mod q^2) and q^(p-1)=1 (mod p^2). The only known examples are (2, 1093), (3, 1006003), (5 , 1645333507), (83, 4871), (911, ...