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Here's a screenshot, in case the equation doesn't render for you.

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Here's a screenshot, in case the equation doesn't render for you.

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The sum of the inverse hyperbolic tangent of 1 over n cubed from n equals 2 to infinity is equal to the inverse hyperbolic tangent of one fifth.

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Does the following equation have a nice geometric interpretation?

\[\sum_{n=2}^{\infty} \tanh^{-1} \frac{1}{n^3} = \tanh^{-1} \frac{1}{5}\]

# orderedItems[].object.contentMap.en
Does the following equation have a nice geometric interpretation?

\[\sum_{n=2}^{\infty} \tanh^{-1} \frac{1}{n^3} = \tanh^{-1} \frac{1}{5}\]

# orderedItems[].object.content
@drScott It's because n! can be factored as 1*2*...*n.

# orderedItems[].object.contentMap.en
@drScott It's because n! can be factored as 1*2*...*n.

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If N is the product of two primes, p < q, then 𝑝=gcd(𝑁,⌊√𝑁⌋!) . This is not a practical factorization method, because we don't know an efficient algorithm to compute factorials mod N. But it makes me wonder if a similar sequence of highly composite numbers could be used as the basis of an efficient algorithm for integer factorization.

# orderedItems[].object.contentMap.en
If N is the product of two primes, p < q, then 𝑝=gcd(𝑁,⌊√𝑁⌋!) . This is not a practical factorization method, because we don't know an efficient algorithm to compute factorials mod N. But it makes me wonder if a similar sequence of highly composite numbers could be used as the basis of an efficient algorithm for integer factorization.

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1, 3, and 11677 are the only odd numbers less than 10^11 whose squares can be written in base 8 using only the digits 0 and 1.

1² = 1₈
3² = 11₈
11677² = 1010111111₈

Are there any other (odd) numbers with this property?

# orderedItems[].object.contentMap.en
1, 3, and 11677 are the only odd numbers less than 10^11 whose squares can be written in base 8 using only the digits 0 and 1.

1² = 1₈
3² = 11₈
11677² = 1010111111₈

Are there any other (odd) numbers with this property?

# orderedItems[].object.content
@christianp This makes me wonder about other kinds of analog clocks. Has anyone ever made a clock whose minute and hour hands move in opposite directions?

# orderedItems[].object.contentMap.en
@christianp This makes me wonder about other kinds of analog clocks. Has anyone ever made a clock whose minute and hour hands move in opposite directions?

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I ate your plums
No time to explain

# orderedItems[].object.contentMap.en
I ate your plums
No time to explain

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@adityakhanna Yes.

# orderedItems[].object.contentMap.en
@adityakhanna Yes.

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Find a sequence of 1000 positive integers in arithmetic progression whose digit sums in base 10 are equal.

# orderedItems[].object.contentMap.en
Find a sequence of 1000 positive integers in arithmetic progression whose digit sums in base 10 are equal.

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@minimaliste13 The expected number of terms is e = 2.71828...

# orderedItems[].object.contentMap.en
@minimaliste13 The expected number of terms is e = 2.71828...

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Is there an algorithm for computing the sum of the decimal digits of 2^n that is more efficient than computing the complete decimal expansion of 2^n?

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Is there an algorithm for computing the sum of the decimal digits of 2^n that is more efficient than computing the complete decimal expansion of 2^n?

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I have a question about Sudokus with "negative clues." A negative clue is a statement that a particular cell does NOT contain a particular value, e.g. "the value in row 3 and column 5 is not 7." What is the minimum number of negative clues that can uniquely determine a completed Sudoku grid?

I posted the question here: https://puzzling.stackexchange.com/que stions/128396/minimum-number-of-negative-clues-for-a-9-x-9-sudoku

# orderedItems[].object.contentMap.en
I have a question about Sudokus with "negative clues." A negative clue is a statement that a particular cell does NOT contain a particular value, e.g. "the value in row 3 and column 5 is not 7." What is the minimum number of negative clues that can uniquely determine a completed Sudoku grid?

I posted the question here: https://puzzling.stackexchange.com/que stions/128396/minimum-number-of-negative-clues-for-a-9-x-9-sudoku