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An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "xRy" to mean (x,y) is an ...

A deeper result than the Hardy-Ramanujan theorem. Let N(x,a,b) be the number of integers in [n,x] such that inequality a<=(omega(n)-lnlnn)/(sqrt(lnlnn))<=b (1) holds, where ...

The sequence of numbers obtained by letting a_1=2, and defining a_n=lpf(1+product_(k=1)^(n-1)a_k) where lpf(n) is the least prime factor. The first few terms are 2, 3, 7, 43, ...

An Euler-Jacobi pseudoprime to a base a is an odd composite number n such that (a,n)=1 and the Jacobi symbol (a/n) satisfies (a/n)=a^((n-1)/2) (mod n) (Guy 1994; but note ...

The numbers 2^npq and 2^nr are an amicable pair if the three integers p = 2^m(2^(n-m)+1)-1 (1) q = 2^n(2^(n-m)+1)-1 (2) r = 2^(n+m)(2^(n-m)+1)^2-1 (3) are all prime numbers ...

A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into ...

If {f_n} is a sequence of nonnegative measurable functions, then intlim inf_(n->infty)f_ndmu<=lim inf_(n->infty)intf_ndmu. (1) An example of a sequence of functions for which ...

Every finite simple group (that is not cyclic) has even group order, and the group order of every finite simple noncommutative group is doubly even, i.e., divisible by 4 ...

The Feller-Tornier constant is the density of integers that have an even number of prime factors p_i^(a_i) with a_1>1 in their prime factorization. It is given by ...

The eigenvector corresponding to the second smallest eigenvalue (i.e., the algebraic connectivity) of the Laplacian matrix of a graph G. The Fiedler vector is used in ...