Is Yaranaikian sequence power-free?

The following sequence $69,696,6969,69696,....$ is called Yaranaikian. To clarify, each term is formed by writing 6 and 9 alternately(starting off with 6 from the left) . It was found that 69696 is actually a perfect square, its square root being 264. That was when my curiosity kicked in. Is there anymore perfect square in the sequence?. After some observation and works, I managed to prove that Yaranaikian sequence is sqaure-free , except for 69696 of course; the proof can be found on my fb's note section,if you're interested. However, it didn't stop there. A more challenging question popped up, as stated below;

Is Yaranaikian sequence excluding 69696 power-free, that is it doesn't contain a term of the form $a^n$ where $a,n$ are positive integers greater than 1?

It's still an open question , but I managed to crack some of the cases as described in the papers on my fb's note section. I sincerely appreciate any attempts or suggestions from anyone.

One may slightly generalise the problem by substituting 6 and 9 by any digits.

Note by ComradeGeneral Poon Thongsai
4 years, 9 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 1

paragraph 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...