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It is possible to perform multiplication of large numbers in (many) fewer operations than the usual brute-force technique of "long multiplication." As discovered by Karatsuba ...

Given a field F and an extension field K superset= F, an element alpha in K is called algebraic over F if it is a root of some nonzero polynomial with coefficients in F. ...

For every dimension n>0, the orthogonal group O(n) is the group of n×n orthogonal matrices. These matrices form a group because they are closed under multiplication and ...

Abstract Algebra

Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. ...

The prime zeta function P(s)=sum_(p)1/(p^s), (1) where the sum is taken over primes is a generalization of the Riemann zeta function zeta(s)=sum_(k=1)^infty1/(k^s), (2) where ...

The logarithm of the reciprocal of a number, equal to the negative of the logarithm of the number itself, cologx=log(1/x)=-logx.

A dual number is a number x+epsilony, where x,y in R and epsilon is a matrix with the property that epsilon^2=0 (such as epsilon=[0 1; 0 0]).

A number D that possesses no common divisor with a prime number p is either a quadratic residue or nonresidue of p, depending whether D^((p-1)/2) is congruent mod p to +/-1.

The area of a rational right triangle cannot be a square number. This statement is equivalent to "a congruum cannot be a square number."