# 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

3 months, 1 week ago

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$\begin{array} ~2 = 4 - 2 = {\color{blue}2^2 - 2^1} \\ 4 = 8 - 4 = {\color{green}2^3 - 2^2} \\ 6 = 8 - 2 = {\color{red}2^3 - 2^1} \\ 8 = 16 - 8 = {\color{pink}2^4 - 2^3} \\ \vdots \\ and~so~on \\ \end{array}$

- 3 months ago

How do you get all of this colouring and stuff

- 3 months ago

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

- 3 months ago

- 3 months ago

Great. Will try them later

- 3 months ago

You are a teacher. Someone who sacrifices

- 3 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.

- 3 months ago

Finish quoting

- 3 months ago

I should quote this

- 3 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.

- 3 months ago

Thank you

- 3 months ago

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

- 3 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

- 3 months ago

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

- 3 months ago

THANK YOU

- 3 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.

- 3 months ago

I know. I am also Indian

- 3 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.

- 3 months ago

Did you see my note : Tricks for memorizing e

- 3 months ago

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

- 3 months ago

Comment deleted 3 months ago

Good.

- 3 months ago

Did you see my about in my profile

- 3 months ago

- 3 months ago

How about the updated one. Like now

- 3 months ago

I worth not be placed among those great people.

- 3 months ago

But you spent the whole day patiently teaching me

- 3 months ago

I regularly visit brilliant as it is a wonderfully website. As a part of my routine I came today and I founded your queries and they are quite interesting so I clarified you. Now I am satisfied that I am able to clear the confusion in a young mind.

- 3 months ago

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.

- 3 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

- 3 months ago