Tag Archives: powers

A combinatorial proof: PIE a la mode!

Posted on October 25, 2019 by Brent

Continuing from my last post in this series, we’re trying to show that , where is defined as which is what we get when we start with a sequence of consecutive th powers and repeatedly take successive differences. Recall that … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged consecutive, difference, function, integers, matching, powers | Comments Off

A combinatorial proof: counting bad functions

Posted on September 29, 2019 by Brent

In a previous post we derived the following expression: . We are trying to show that , in order to show that starting with a sequence of consecutive th powers and repeatedly taking successive differences will always result in . … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged consecutive, difference, function, integers, matching, powers | 1 Comment

A combinatorial proof: functions and matchings

Posted on September 26, 2019 by Brent

We’re trying to prove the following equality (see my previous post for a recap of the story so far): In particular we’re trying to show that the two sides of this equation correspond to two different ways to count the … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged consecutive, difference, function, integers, matching, powers | 5 Comments

A combinatorial proof: the story so far

Posted on September 6, 2019 by Brent

In my last post I reintroduced this seemingly odd phenomenon: Start with consecutive integers and raise them all to the th power. Then repeatedly take pairwise differences (i.e. subtract the first from the second, and the second from the third, … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged consecutive, difference, integers, powers | 1 Comment

A combinatorial proof: reboot!

Posted on August 31, 2019 by Brent

More than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, you always end up with , … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged consecutive, difference, integers, powers | 11 Comments

The curious powers of 1 + sqrt 2: recurrences

Posted on July 6, 2017 by Brent

In my previous post, we found an answer to the question: What’s the 99th digit to the right of the decimal point in the decimal expansion of ? However, the solution depended on having the clever idea to add . … Continue reading →

Posted in number theory, puzzles | Tagged conjecture, digits, powers, puzzle, ratio, silver | 8 Comments

The curious powers of 1 + sqrt 2: a clever solution

Posted on June 21, 2017 by Brent

Recall that we are trying to answer the question: What’s the 99th digit to the right of the decimal point in the decimal expansion of ? In my previous post, we computed for some small and conjectured that the answer … Continue reading →

Posted in number theory, puzzles | Tagged conjecture, digits, powers, puzzle, ratio, silver | 2 Comments

The curious powers of 1 + sqrt 2: conjecture

Posted on June 17, 2017 by Brent

In my previous post I related the following puzzle from Colin Wright: What’s the 99th digit to the right of the decimal point in the decimal expansion of ? Let’s play around with this a bit and see if we … Continue reading →

Posted in number theory, puzzles | Tagged conjecture, digits, powers, puzzle, ratio, silver | 3 Comments

The curious powers of 1 + sqrt 2

Posted on June 6, 2017 by Brent

Recently on mathstodon.xyz, Colin Wright posted the following puzzle: What’s the 99th digit to the right of the decimal point in the decimal expansion of ? Of course, it’s simple enough to use a computer to find the answer; any … Continue reading →

Posted in challenges, number theory, puzzles | Tagged powers, puzzle, ratio, silver | 15 Comments

Differences of powers of consecutive integers, part II

Posted on February 16, 2012 by Brent

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