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Any composite number n with p|(n/p-1) for all prime divisors p of n. n is a Giuga number iff sum_(k=1)^(n-1)k^(phi(n))=-1 (mod n) (1) where phi is the totient function and ...

Euler (1738, 1753) considered the series s_a(x)=sum_(n=1)^infty[1/(1-a^n)product_(k=0)^(n-1)(1-xa^(-k))]. He showed that just like log_a(a^n)=n, s_a(a^n)=n for nonnegative ...

Let f be an entire function of finite order lambda and {a_j} the zeros of f, listed with multiplicity, then the rank p of f is defined as the least positive integer such that ...

The second solution Q_l(x) to the Legendre differential equation. The Legendre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials. ...

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates theta and phi from uniform distributions theta in [0,2pi) and phi in ...

The value for zeta(2)=sum_(k=1)^infty1/(k^2) (1) can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and ...

Let L_n be the n×n matrix whose (i,j)th entry is 1 if j divides i and 0 otherwise, let Phi_n be the n×n diagonal matrix diag(phi(1),phi(2),...,phi(n)), where phi(n) is the ...

Let A={a_1,a_2,...} be a free Abelian semigroup, where a_1 is the identity element, and let mu(n) be the Möbius function. Define mu(a_n) on the elements of the semigroup ...

The term energy has an important physical meaning in physics and is an extremely useful concept. There are several forms energy defined in mathematics. In measure theory, let ...

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 ...