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The second theorem of Mertens states that the asymptotic form of the harmonic series for the sum of reciprocal primes is given by sum_(p<=x)1/p=lnlnx+B_1+o(1), where p is a ...

A polynomial admitting a multiplicative inverse. In the polynomial ring R[x], where R is an integral domain, the invertible polynomials are precisely the constant polynomials ...

An addition-multiplication square is a square of integers that is simultaneously a magic square and multiplication magic square. In 1955, Horner found a square of order eight ...

A shift-invariant operator Q for which Qx is a nonzero constant. 1. Qa=0 for every constant a. 2. If p(x) is a polynomial of degree n, Qp(x) is a polynomial of degree n-1. 3. ...

The series producing Brun's constant converges even if there are an infinite number of twin primes, first proved by Brun (1919).

If f(x) is positive and decreases to 0, then an Euler constant gamma_f=lim_(n->infty)[sum_(k=1)^nf(k)-int_1^nf(x)dx] can be defined. For example, if f(x)=1/x, then ...

Let s(n)=sigma(n)-n, where sigma(n) is the divisor function and s(n) is the restricted divisor function. Then the sequence of numbers s^0(n)=n,s^1(n)=s(n),s^2(n)=s(s(n)),... ...

Any entire analytic function whose range omits two points must be a constant function. Of course, an entire function that omits a single point from its range need not be a ...

A valuation for which |x|<=1 implies |1+x|<=C for the constant C=1 (independent of x). Such a valuation does not satisfy the strong triangle inequality |x+y|<=max(|x|,|y|).

The geodesics on a general cylinder generated by lines parallel to a line l with which the tangent makes a constant angle.