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As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exist a general algorithm for solving a general quartic ...

An n-step Fibonacci sequence {F_k^((n))}_(k=1)^infty is defined by letting F_k^((n))=0 for k<=0, F_1^((n))=F_2^((n))=1, and other terms according to the linear recurrence ...

Closed forms are known for the sums of reciprocals of even-indexed Fibonacci numbers P_F^((e)) = sum_(n=1)^(infty)1/(F_(2n)) (1) = ...

The Euler polynomial E_n(x) is given by the Appell sequence with g(t)=1/2(e^t+1), (1) giving the generating function (2e^(xt))/(e^t+1)=sum_(n=0)^inftyE_n(x)(t^n)/(n!). (2) ...

Newton's forward difference formula is a finite difference identity giving an interpolated value between tabulated points {f_p} in terms of the first value f_0 and the powers ...

nu_((r))=sum_(x)x^((r))f(x), where x^((r))=x(x-1)...(x-r+1).

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For ...

The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The ...

A pair of primes (p,q) that sum to an even integer 2n=p+q are known as a Goldbach partition (Oliveira e Silva). Letting r(2n) denote the number of Goldbach partitions of 2n ...

An edge-magic graph is a labeled graph with e graph edges labeled with distinct elements {1,2,...,e} so that the sum of the graph edge labels at each graph vertex is the ...