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Pythagoras's theorem states that the diagonal d of a square with sides of integral length s cannot be rational. Assume d/s is rational and equal to p/q where p and q are ...

If there is an integer 0<x<p such that x^2=q (mod p), (1) i.e., the congruence (1) has a solution, then q is said to be a quadratic residue (mod p). Note that the trivial ...

The nth Ramanujan prime is the smallest number R_n such that pi(x)-pi(x/2)>=n for all x>=R_n, where pi(x) is the prime counting function. In other words, there are at least n ...

The sum c_q(m)=sum_(h^*(q))e^(2piihm/q), (1) where h runs through the residues relatively prime to q, which is important in the representation of numbers by the sums of ...

A number n is said to be refactorable, sometimes also called a tau number (Kennedy and Cooper 1990), if it is divisible by the number of its divisors sigma_0(n), where ...

A usually simple algorithm or identity. The term is frequently applied to specific orders of Newton-Cotes formulas. The designation "rule n" is also given to the nth ...

A partition whose conjugate partition is equivalent to itself. The Ferrers diagrams corresponding to the self-conjugate partitions for 3<=n<=10 are illustrated above. The ...

A sequence is an ordered set of mathematical objects. Sequences of object are most commonly denoted using braces. For example, the symbol {2n}_(n=1)^infty denotes the ...

A series is an infinite ordered set of terms combined together by the addition operator. The term "infinite series" is sometimes used to emphasize the fact that series ...

Let the values of a function f(x) be tabulated at points x_i equally spaced by h=x_(i+1)-x_i, so f_1=f(x_1), f_2=f(x_2), ..., f_(11)=f(x_(11)). Then Shovelton's rule ...