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A number which is simultaneously square and triangular. Let T_n denote the nth triangular number and S_m the mth square number, then a number which is both triangular and ...

There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron ...

A congruent number can be defined as an integer that is equal to the area of a rational right triangle (Koblitz 1993). Numbers (a,x,y,z,t) such that {x^2+ay^2=z^2; ...

Consider solutions to the equation x^y=y^x. (1) Real solutions are given by x=y for x,y>0, together with the solution of (lny)/y=(lnx)/x, (2) which is given by ...

As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers. ...

An idoneal number, also called a suitable number or convenient number, is a positive integer D for which the fact that a number is a monomorph (i.e., is expressible in only ...

The second-order ordinary differential equation (1-x^2)y^('')-2(mu+1)xy^'+(nu-mu)(nu+mu+1)y=0 (1) sometimes called the hyperspherical differential equation (Iyanaga and ...

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, ...

A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a ...