A Question Inspired by Pi Han Goh

Can there be more than 5 positive integers such that they are in a harmonic progression ie there reciprocals are in arithmetic progression? If yes, the find some with more than 5 terms and show your working, and if not, prove your observation.

Note by Kushagra Sahni
2 years, 4 months ago

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Note that $$\frac{1}{n!}, \frac{2}{n!}, \frac{3}{n!}, \ldots, \frac{n}{n!}$$ form an arithmetic progression, and the numerator divides the denominator so they all simplify to unit fractions. Thus their reciprocals are positive integers that form a harmonic progression. Take $$n$$ as large as you want.

- 2 years, 3 months ago

That's a cool solution. Do other solutions exist?

- 2 years, 3 months ago

Of course. Let $$a_1, a_2, \ldots, a_n$$ be an arithmetic progression of positive integers, and let $$P$$ be their least common multiple. Then $$\frac{kP}{a_1}, \frac{kP}{a_2}, \ldots, \frac{kP}{a_n}$$ is a solution, for any positive integer $$k$$. (The above is when you put $$a_i = i$$ and $$k$$ is such so $$kP = n!$$.)

Whether the above gives all the solutions, I haven't proved it yet, although I think it does.

- 2 years, 3 months ago

Yes I guess that covers all.

- 2 years, 3 months ago