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

Note by Mohmmad Farhan
3 months, 1 week ago

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

Ram Mohith - 3 months ago

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How do you get all of this colouring and stuff

Mohmmad Farhan - 3 months ago

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

Ram Mohith - 3 months ago

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@Ram Mohith Great. Will try them later

Mohmmad Farhan - 3 months ago

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@Ram Mohith You are a teacher. Someone who sacrifices

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan 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.

Ram Mohith - 3 months ago

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@Ram Mohith Finish quoting

Mohmmad Farhan - 3 months ago

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@Ram Mohith I should quote this

Mohmmad Farhan - 3 months ago

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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.

Ram Mohith - 3 months ago

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@Ram Mohith Thank you

Mohmmad Farhan - 3 months ago

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

Mohmmad Farhan - 3 months ago

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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

Mohmmad Farhan - 3 months ago

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

Ram Mohith - 3 months ago

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@Ram Mohith THANK YOU

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan 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.

Ram Mohith - 3 months ago

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@Ram Mohith I know. I am also Indian

Mohmmad Farhan - 3 months ago

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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.

Ram Mohith - 3 months ago

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@Ram Mohith Did you see my note : Tricks for memorizing e

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan Yes. I have seen it. It is a good observation.

Ram Mohith - 3 months ago

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Comment deleted 3 months ago

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@Mohmmad Farhan Good.

Ram Mohith - 3 months ago

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@Ram Mohith Did you see my about in my profile

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan Yes. About your idols.

Ram Mohith - 3 months ago

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@Ram Mohith How about the updated one. Like now

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan I worth not be placed among those great people.

Ram Mohith - 3 months ago

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@Ram Mohith But you spent the whole day patiently teaching me

Mohmmad Farhan - 3 months ago

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@Mohmmad Farhan 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.

Ram Mohith - 3 months 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.

Ram Mohith - 3 months ago

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

Mohmmad Farhan - 3 months ago

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