# Instructive Examples in Kraitchik’s Method

While discussing the history of the modern factoring, Carl Pomerance’s 1996 expository piece “A Tale of Two Sieves” describes a factoring algorithm called Kraitchik’s Method and demonstrates the algorithm by factoring 2041.

The example is nice; certainly nicer and more illustrative than what you might produce at random. But exactly how special is Pomerance’s 2041 example?

# A Proof of Stirling’s Approximation via Contour Integration

Stirling’s approximation gives a useful estimate for large factorials. This post contains a (new?) proof of Stirling’s formula which relies on properties of the Riemann zeta function.

# Fractal Sets and Arithmetic Progressions

If a set of positive integers contains no arithmetic progressions, how large can it be? In this post, we study this question in the context of harmonic sums.

# On Unparsed Substitution Ciphers

This post discusses the problem of computer-assisted decryption of un-parsed substitution ciphers. Sample Mathematica code is linked within.

# Using Groups to Factor Integers

Most factorization algorithms in use today fit in one of two camps: sieve-based methods based on congruences of squares, and algorithms based on decompositions of algebraic groups. In this article, we trace the common thread connecting the latter.