Waste less time on Facebook — follow Brilliant.
×

The prime next door

Let us define \(Z(n)\) , such that for the input of natural \(n\) , \(Z(n)\) gives the immediate prime number next to \(n\) .

Then \[ a^{Z(n)} \equiv a \mod n \]


=> \(a\) and \(n\) belong to the set of natural numbers where \(n\) is not a prime number and greater than one ..

Note by Chinmay Sangawadekar
1 year, 2 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

The claim is not true.

For example, with \( n = 32, Z(n) = 37 \), we have \( 3^{37} \equiv 19 \pmod{32} \).

I believe what happened was that you were testing small cases where the prime factors of \(n\) were small and distinct, which combined with euler's theorem led to this being true in several cases. As such, I went with a prime power, and then used one that was large enough.

Calvin Lin Staff - 1 year, 2 months ago

Log in to reply

Can we find range if solutions for this ?

Chinmay Sangawadekar - 1 year, 2 months ago

Log in to reply

Awesome result! Is it original?

I'll try proving it.

Harsh Shrivastava - 1 year, 2 months ago

Log in to reply

Yeah it is , but it is disproved for large values :( .

Chinmay Sangawadekar - 1 year, 2 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...