The previous post looked at Euler’s method for accelerating the convergence of a slowly converging alternating series. Both hypotheses are necessary. The signs must alternate between terms, and applying the method to a series that is already converging quickly can slow down convergence. This post looks at Aitken’s method for speeding up the convergence of […]

# Category: Math

## Accelerating an alternating series

The most direct way of computing the sum of an alternating series, simply computing the partial sums in the terms get small enough, may not be the most efficient. Euler figured this out in the 18th century. For our demo we’ll evaluate the Struve function defined by the series Note that the the terms in […]

## Beating the odds on the Diffie-Hellman decision problem

There are a couple variations on the Diffie-Hellman problem in cryptography: the computation problem (CDH) and the decision problem (DDH). This post will explain both and give an example of where the former is hard and the latter easy. The Diffie-Hellman problems The Diffie-Hellman problems are formulated for an Abelian group. The main group we […]

## Magic square links and errata

Someone pointed out that what I called a knight’s tour magic square is technically a semi-magic square: the rows and columns add up to the same constant, but the diagonals do not. It turns out there are no strict magic squares formed by knight’s tours. This was proved in 2003. See a news article here. […]

## Quaternion reference in the Vulgate

To contemporary ears “quaternion” refers to a number system discovered in the 19h century, but there were a couple precedents. Both refer to something related to a group of four things, but there is no relation to mathematical quaternions other than that they have four dimensions. I’ve written before about Milton’s use of the word […]

## Fame, difficulty, and usefulness

Pierre Fermat is best known for two theorems, dubbed his “last” theorem and his “little” theorem. His last theorem is famous, difficult to prove, and useless. His little theorem is relatively arcane, easy to prove, and extremely useful. There’s little relation between technical difficulty and usefulness. Fermat’s last theorem Fermat’s last theorem says there are […]

## Twisted elliptic curves

This morning I was sitting at a little bakery thinking about what to do before I check out of my hotel. I saw that the name of the bakery was Twist Bakery & Cafe, and that made me think of writing about twisted elliptic curves when I got back to my room. Twist of an […]

## Integral approximation trick

Here’s a simple integration approximation that works remarkably well in some contexts. Suppose you have an integrand that looks roughly like a normal density, something with a single peak that drops off fairly quickly on either side of the peak. The majority of integrals that arise in basic applications of probability and statistics fit this […]

## Number of feet in a mile

Here are a couple amusing things I’ve run across recently regarding the number of feet in a mile. Both are frivolous, but also have a more serious side. Mnemonic First, you can use “five tomatoes” as a mnemonic for remembering that there are 5280 feet in a mile. “Five tomatoes” is a mnemonic for the […]

## Discriminant of a cubic

The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² – 4ac. If the discriminant Δ is zero, the equation has a double root, i.e. there is a unique x that makes the equation zero, and it counts twice as a root. If the discriminant is not zero, […]

## Distribution of quadratic residues

Let p be an odd prime number. If the equation x² = n mod p has a solution then n is a square mod p, or in classical terminology, n is a quadratic residue mod p. Half of the numbers between 0 and p are quadratic residues and half are not. The residues are distributed […]

## Serious applications of a party trick

In a group of 30 people, it’s likely that two people have the same birthday. For a group of 23 the probability is about 1/2, and it goes up as the group gets larger. In a group of a few dozen people, it’s unlikely that anyone will have a particular birthday, but it’s likely that […]

## Symmetry in exponential sums

Today’s exponential sum is highly symmetric: These sums are often symmetric, but not always. For example, here’s the sum from a couple days ago: It’s not obvious from looking at the parameters whether a sum will be symmetric or not. Maybe someone could find a prove criteria for a sum to have certain symmetries. For […]

## Proto-calculus

David Bressoud has written a new book entitled Calculus Reordered: A History of the Big Ideas. He presents the major themes of calculus in historical order, which is very different from the order in which it is now taught. We now begin with limits, then differentiation, integration, and infinite series. Historically, integration came first and […]

## Homomorphic encryption

A function that satisfies f(x*y) = f(x)*f(y) is called a homomorphism. The symbol “*” can stand for any operation, and it need not stand for the same thing on both sides of the equation. Technically * is the group operation, and if the function f maps elements of one group to another, the group operation […]

## Category theory for data science: cautious optimism

I’m cautiously optimistic about applications of category theory. I believe category theory has the potential to be useful, but I’m skeptical of most claims to have found useful applications. Category theory has found useful application, especially behind the scenes, but a lot of supposed applications remind me of a line from Colin McLarty: [Jean-Pierre] Serre […]

## Software to factor integers

In my previous post, I showed how changing one bit of a semiprime (i.e. the product of two primes) creates an integer that can be factored much faster. I started writing that post using Python with SymPy, but moved to Mathematica because factoring took too long. SymPy vs Mathematica When I’m working in Python, SymPy […]

## Making public keys factorable with Rowhammer

The security of RSA encryption depends on the fact that the product of two large primes is difficult to factor. So if p and q are large primes, say 2048 bits each, then you can publish n = pq with little fear that someone can factor n to recover p and q. But if you […]

## Bounds on the nth prime

The nth prime is approximately n log n. For more precise estimates, there are numerous upper and lower bounds for the nth prime, each tighter over some intervals than others. Here I want to point out upper and lower bounds from a dissertation by Christian Axler on page viii. First, define Then for sufficiently large […]

## Converting between nines and sigmas

Nines and sigmas are two ways to measure quality. You’ll hear something has four or five nines of reliability or that some failure is a five sigma event. What do these mean, and how do you convert between them? Definitions If a system has fives nines of availability, that means the probability of the system […]