# 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

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

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