Quadratic Reciprocity and the Theta Function

One of the better-known proofs of quadratic reciprocity involves the Gauss sums. This post gives a variant proof which motivates the introduction of Gauss sums using the Jacobi theta function.

lovely-lucky-lambs (foobar)

This post discusses one of the coding challenges from Google’s foobar program, lovely-lucky-lambs. More specifically, it discusses an error in the official solution to this problem and how this may have occurred.

Statistical Attacks on the Autokey Cipher

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

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.

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.