# Interesting prime powers relationship

One day, I was thinking about primes that add up to 30 and all I thought about was:

$(7,23) , (11,19) , (13,17)$

And I found out that the interesting thing when I got the difference of each pair they were actually powers of 2!

$23-7=16=4^2=2^4$

$19-11=8=2^3$

$17-13=4=2^2$

And the prime factorization of 30 was $2^1$ times $3^1$ times $5^1$ which were consecutive primes. So I suspect there is a relationship that does this

2 years, 11 months ago

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.

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:

I think the probable reason may that any multiple of $2$ can be expressed in the form $2^a - 2^b$ where $a,b$ are integers. Also, when any two prime numbers except 2 are added or subtracted they always result in an even number.

- 2 years, 11 months ago

$\begin{array}{c}~2 = 4 - 2 = {\color{#3D99F6}2^2 - 2^1} \\ 4 = 8 - 4 = {\color{#20A900}2^3 - 2^2} \\ 6 = 8 - 2 = {\color{#D61F06}2^3 - 2^1} \\ 8 = 16 - 8 = {\color{#E81990}2^4 - 2^3} \\ \vdots \\ and~so~on \\ \end{array}$

- 2 years, 11 months ago

Thank you. Now I have to give you a treat for giving me this knowledge

- 2 years, 11 months ago

No problem. I am surprised to see that a young boy of age just $10$ years is having this much curiosity about learning the subject.

- 2 years, 11 months ago

Did you see my note : Tricks for memorizing e

- 2 years, 11 months ago

Yes. I have seen it. It is a good observation.

- 2 years, 11 months ago

Please stop. Do you know how irritating it is when someone in your class calls you a genius. I am being bullied everyday by people who positively give me complements

- 2 years, 11 months ago

No it is just a reward from a person who is just elder than you in knowledge and experience.

- 2 years, 11 months ago

THANK YOU

- 2 years, 11 months ago

It's Ok. I liked your curiosity of gaining and grasping the subject which I used to have in my childhood. That's the reason my fundas are strong.

Note: fundas means fundamentals.

- 2 years, 11 months ago

I know. I am also Indian

- 2 years, 11 months ago

How do you get all of this colouring and stuff

- 2 years, 11 months ago

Using latex codes. You can move you cursor on the text and you will be able to see the code or just click "Toggle Latex" which will appear in the section where profile, stats,account settings are there. Check it.

- 2 years, 11 months ago

Thank you

- 2 years, 11 months ago

If you want I will give you latex codes links wait.

- 2 years, 11 months ago

You are a teacher. Someone who sacrifices

- 2 years, 11 months ago

Don't use such big words. I am still just a student. When compared to the knowledge of highly intelligent members in brilliant my intelligence is like a pebble. But the thing is I made my fundamentals strong. Those whose fundamentals are strong they need not see back in their life.

- 2 years, 11 months ago

I should quote this

- 2 years, 11 months ago

Finish quoting

- 2 years, 11 months ago

- 2 years, 11 months ago

Great. Will try them later

- 2 years, 11 months ago

For 210 I got 173,37 and the difference was 136 which was equal to 10^2+6^2=2(6^2)+8^2

- 2 years, 11 months ago