# Statistical Attacks on the Autokey Cipher

Letter frequency analysis is a common technique for solving substitution and Vigenère ciphers. In this note, we apply similar techniques to the autokey cipher. This isn’t much harder but appears to be less well-known.

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# A Proof of the Infinitude of Congruent Numbers via Dirichlet Series

Euler gave a proof of the infinitude of primes which used the meromorphic behavior of the Riemann zeta function. In this post, we show that similar ideas can be used to show the infinitude of congruent numbers.

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# 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?

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# 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.

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# 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.

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# On Unparsed Substitution Ciphers

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

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# 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.

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# The Forgotten Method of Prosthaphaeresis

In this post, I discuss the method of prosthaphaeresis, a proto-logarithm that enabled celestial navigation for the quarter-century predating the introduction of Napier’s logarithm.

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# Integrating Secant

The integral formula for secant is often introduced in a very artificial way. In this post, I look at the history of this integral and give several derivations.

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# Large Prime Factors of a Quadratic Polynomial

One of Landau’s four problems from 1912 concerns the infinitude of primes in the values of a certain quadratic polynomial.

In this post, we show that the largest prime factors of the values of this polynomial are “relatively large” infinitely often.

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