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# Tag Archives: powers

## The curious powers of 1 + sqrt 2: recurrences

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

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

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

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

## 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

## Differences of powers of consecutive integers

Patrick Vennebush of Math Jokes 4 Mathy Folks recently wrote about the following procedure that yields surprising results. Choose some positive integer . Now, starting with consecutive integers, raise each integer to the th power. Then take pairwise differences by … Continue reading

Posted in arithmetic, pattern
Tagged consecutive, difference, integers, powers, surprising
16 Comments