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

Easy Math Editor

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Comments

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 - 2 years, 10 months ago

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

Ram Mohith - 2 years, 10 months ago

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