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Let P=a_1x+a_2x^2+... be an almost unit in the integral domain of formal power series (with a_1!=0) and define P^k=sum_(n=k)^inftya_n^((k))x^n (1) for k=+/-1, +/-2, .... If ...

Let (K,|·|) be a complete non-Archimedean valuated field, with valuation ring R, and let f(X) be a power series with coefficients in R. Suppose at least one of the ...

A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c. By the Pythagorean theorem, this is ...

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

Solve the Pell equation x^2-92y^2=1 in integers. The smallest solution is x=1151, y=120.

Calculating the 13th root of a large number (that is a perfect 13th power) is a famous mental calculation challenge. However, because of difficulties in standardizing the ...

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies (a+bi)^(c+di)=(a^2+b^2)^((c+id)/2)e^(i(c+id)arg(a+ib)), ...

The theory of classifying problems based on how difficult they are to solve. A problem is assigned to the P-problem (polynomial-time) class if the number of steps needed to ...

The kth exterior power of an element alpha in an exterior algebra LambdaV is given by the wedge product of alpha with itself k times. Note that if alpha has odd degree, then ...

The falling factorial (x)_n, sometimes also denoted x^(n__) (Graham et al. 1994, p. 48), is defined by (x)_n=x(x-1)...(x-(n-1)) (1) for n>=0. Is also known as the binomial ...