# 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

7 months, 2 weeks ago

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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.

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week 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.

- 7 months, 1 week ago

Did you see my note : Tricks for memorizing e

- 7 months, 1 week ago

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

- 7 months, 1 week 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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

THANK YOU

- 7 months, 1 week 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.

- 7 months, 1 week ago

I know. I am also Indian

- 7 months, 1 week ago

How do you get all of this colouring and stuff

- 7 months, 1 week 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.

- 7 months, 1 week ago

Thank you

- 7 months, 1 week ago

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

- 7 months, 1 week ago

You are a teacher. Someone who sacrifices

- 7 months, 1 week 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.

- 7 months, 1 week ago

I should quote this

- 7 months, 1 week ago

Finish quoting

- 7 months, 1 week ago

- 7 months, 1 week ago

Great. Will try them later

- 7 months, 1 week 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

- 7 months, 1 week ago