Bertrand's postulate and subgroup growth

Bertrand ’ s postulate and subgroup growth

In this article we investigate the L1–norm of certain functions on groups called divisibility functions. Using these functions, their connection to residual finiteness, and integration theory on profinite groups, we define the residual average of a finitely generated group. One of the main results in this article is the finiteness of residual averages on finitely generated linear groups. Whethe...

Bertrand's postulate

Bertrand’s postulate is an early result on the distribution of prime numbers: For every positive integer n, there exists a prime number that lies strictly between n and 2n. The proof is ported from John Harrison’s formalisation in HOL Light [1]. It proceeds by first showing that the property is true for all n greater than or equal to 600 and then showing that it also holds for all n below 600 b...

Pocklington’s Theorem and Bertrand’s Postulate

The following propositions are true: (1) For all real numbers r, s such that 0 ≤ r and s · s < r · r holds s < r. (2) For all real numbers r, s such that 1 < r and r · r ≤ s holds r < s. (3) For all natural numbers a, n such that a > 1 holds an > n. (4) For all natural numbers n, k, m such that k ≤ n and m = b n2 c holds (n m ) ≥ (n k ) . (5) For all natural numbers n, m such that m = b n2 c an...

New Lower Bounds on Subgroup Growth and Homology Growth

Subgroup growth is an important new area of group theory. It attempts to quantify the number of finite index subgroups of a group, as a function of their index. In this paper, we will provide new, strong lower bounds on the subgroup growth of a variety of different groups. This will include the fundamental groups of all finite-volume hyperbolic 3-manifolds. By using the correspondence between s...

Ramanujan Primes and Bertrand's Postulate