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

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

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How do you get all of this colouring and stuff

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

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Make your solutions more attractive

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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.

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

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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

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Note:fundas means fundamentals.Log in to reply

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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.

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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.

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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

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