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# Category Archives: algebra

## A computer-checked proof of the odd order theorem

Big news: a proof of the Feit-Thompson Theorem (also known as the “odd order theorem”) has been completely formalized and verified by a computer, using the Coq proof assistant! Wait, what? Huh? you’re probably thinking. Well, let me unpack that … Continue reading

## Sigmas and sums of squares

Commenter Rachel recently asked, How would you find the sum of ? See here for an explanation of sigma notation—in this case it denotes the sum Of course, for any particular value of we can just plug in values and … Continue reading

## Cassini’s identity

My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading

## A Fibonacci pattern

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences
Tagged fibonacci, number, pattern
4 Comments

## Irrationality of pi: the integral that wasn't

And now for the punchline! Today we’ll show that, for large enough values of , completing the proof of the irrationality of . First, let’s show that is positive when . We know that is positive for . But I … Continue reading

Posted in algebra, calculus, convergence, famous numbers, proof, trig
Tagged inequality, integral, irrational, Niven, pi
8 Comments

## Square roots with pencil and paper: method 2

A little while ago I wrote about the Babylonian method for approximating square roots with pencil and paper. In that post I noted that the Babylonian method is quite efficient, but annoying in some ways since it makes you deal … Continue reading

## Square roots with pencil and paper: the Babylonian method

Everyone knows how to add, subtract, multiply and divide with pencil and paper; but do you know how to find square roots without a calculator? (Incidentally, I highly recommend reading The Feeling of Power by Isaac Asimov, a short story … Continue reading

Posted in algebra, computation, convergence, iteration, number theory
Tagged Babylonian, method, pencil and paper, square root
10 Comments

## Rational and irrational numbers

Most readers of this blog probably know what a rational number is: it’s a number that can be represented as a ratio (hence rational) of two integers. In other words, a fraction. Examples are 3/4, 99/2, and -20837/231, and even … Continue reading

## Perfect numbers, part III

This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the … Continue reading

Posted in algebra, number theory, open problems, primes, solutions
11 Comments