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# Category Archives: iteration

## Differences of powers of consecutive integers, part II

If you spent some time playing around with the procedure from Differences of powers of consecutive integers (namely, raise consecutive integers to the th power, and repeatedly take pairwise differences until reaching a single number) you probably noticed the curious … Continue reading

Posted in arithmetic, iteration, pascal's triangle
Tagged binomial coefficients, consecutive, difference, integers, powers
3 Comments

## u-tube

[This is the eighth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading

Posted in computation, convergence, infinity, iteration, modular arithmetic, number theory, programming
Tagged decadic, Haskell, idempotent, streaming, u
2 Comments

## A self-square number

[This is the seventh in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading

Posted in arithmetic, infinity, iteration, modular arithmetic, proof
Tagged decadic, idempotent, self, square
12 Comments

## Fun with repunit divisors: more solutions

In Fun with repunit divisors I posed the following challenge: Prove that every prime other than 2 or 5 is a divisor of some repunit. In other words, if you make a list of the prime factorizations of repunits, every … Continue reading

Posted in arithmetic, iteration, modular arithmetic, number theory, primes, programming, proof, solutions
Tagged repunit
Comments Off on Fun with repunit divisors: more solutions

## Fun with repunit divisors: proofs

As promised, here are some solutions to the repunit puzzle posed in my previous post. (Stop reading now if you don’t want to see solutions yet!) Prove that every prime other than 2 or 5 is a divisor of some … Continue reading

Posted in iteration, modular arithmetic, number theory, pattern, primes, proof
Tagged divisibility, Fermat, prime, proof, repunit
1 Comment

## m-bracelets

It is easy to generalize number bracelets to moduli other than 10—at each step, add the two previous numbers and take the remainder of the result when divided by m. Here are some pretty pictures I made of the resulting … Continue reading

Posted in arithmetic, fibonacci, iteration, pattern, pictures, sequences
6 Comments

## Number bracelets

Recently I’ve been volunteering with the middle school math club at Penn Alexander, a PreK-8 school in my neighborhood. Today we did (among other things) a fun activity I’d never seen before, called “number bracelets”. The students seemed to enjoy … Continue reading

Posted in arithmetic, iteration, pattern, sequences, teaching
Tagged activity, bracelets, number, Penn Alexander
12 Comments

## The hyperbinary sequence and the Calkin-Wilf tree

And now, the amazing conclusion to this series of posts on Neil Calkin and Herbert Wilf’s paper, Recounting the Rationals, and the answers to all the questions about the hyperbinary sequence. Hold on to your hats! The Calkin-Wilf Tree First, … Continue reading

Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions
Tagged algorithm, binary, Calkin-Wilf, Euclidean, Haskell, hyperbinary, tree
6 Comments

## 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 numbers and decimal expansions

As you may remember from school, rational numbers have a terminating or eventually repeating (periodic) decimal expansion, whereas irrational numbers don’t. So, for example, 0.123123123123…, with 123 repeating forever, is rational (in fact, it is equal to 41/333), whereas something … Continue reading