Zeta and Egyptian Fractions

What values of ξ(n) \xi (-n)\quad (where n = odd number i.e, 1, 3, 5, 7) ..... can not be represented by Egyptian fractions or Egyptian fractions - x (where x is any integer) form. I am only considering Egyptian fraction with length greater than 1.

  • For instance ξ(1) \xi (-1)\quad = -1/12 = [(1/2)+(1/3)+(1/12)] - 1
  • ξ(17) \xi (-17)\quad = 1/2+1/3+1/9+1/625+1/816137+1/1046695702500 - 4
  • ξ(11) \xi (-11)\quad = 691/32760 = (1/48)+(1/3855)+(1/16838640)

However if you see ξ(3) \xi (-3)\quad = 1/120 or ξ(7) \xi (-7)\quad = 1/240 they can not be represented as Egyptian faction of length > 1 Also note that since we are only considering Egyptian fraction with length greater than 1 ξ(33) \xi (-33)\quad which equals (1/12) -12635724796 also becomes part of this sequence

So If we start creating a sequence S(n) for which this is not possible then 3, 7 and 33 are first few candidates. I would like to learn more about this sequence. Also is there any way to know, for what values of n, ξ(n) \xi (-n)\quad will be positive?

Note by Vikram Pandya
5 years, 11 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}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...