Question

as the sum of the exponential numbers of the numbers two and three, all prime numbers greater than three can be written ?
ex. 2+3=5 , 2^2 +3 = 7 , 2 +3^2 = 11 ,2^2 +3^2 = 13,
2^3 +3^2 = 17 ,2^4 +3 = 19 , 2^2 +2^4 +3 = 23 ,...
. how can be this proven ?

Answer 1

@AZ

Answer 2

@angle

Answer 3

\(\color{#0cbb34}{\text{Originally Posted by}}\) @snowflake0531
@AZ
\(\color{#0cbb34}{\text{End of Quote}}\)
any idea ? opinions ??? ty.

Answer 4

like an idea :

2^n = always an even number
3^n = always an od number

2^n + 3^n = always an od number >= 5

Answer 5

or an other idea

we know that every primes greater or equal 5 are the form of 6n +/-1
so

2^n +3^n is posibil have always this form of 6n +/-1 ?

Answer 6

@AZ any idea here pls. ? or opinion ? ty.

Answer 7

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Florisalreadytaken
I am kinda confused — not sure what you are asking for; is your question that why every prime number can be writen as the sum of 2 and 3 in exponential form?
\(\color{#0cbb34}{\text{End of Quote}}\)
,,why" no not is this my question

Answer 8

ok - ty - i know this - but my posted question is
how can you prove that every primes greater then 3 can be written like a sum of 2 and 3 exponential numbers ? look on my examples

Answer 9

Is this a question, or are you asking for a opinion?

Answer 10

every two

Answer 11

@hero

Answer 12

I'm not sure exactly on the approach to this specific problem
but I do know of an allegory where any number can be represented in mod 2 because 2^k and 2^0=1 sum combinations can be used to write any number.
For example 23 = 2^4 + 2^2 + 2^1 + 2^0

I guess this mod idea can be applied to this question in terms of mod 6
The idea being that you only need combinations of 2 and 3 to make 6 and can therefore make any number mod 6

Answer 13

I'm not too good at explaining this thought, sorry

Answer 14

ok @Angle so then i understand your words right so you said any numbers has a form of mod6 - true ?

Answer 15

exactly

every number can be represented as a form of mod 6

0 mod 6
1 mod 6
2 mod 6
3 mod 6
4 mod 6
5 mod 6

Answer 16

every form of mod 6 can also be created by adding a combination of 2 mod 6 and 3 mod 6

Answer 17

ok but i ve asked about how can you prove that every primes can be expressed like a sum of the exponential numbers of 2 and 3

Answer 18

ok primes greater then 3

Answer 19

I am also slightly confused about the restrictions on your question

you included the example of: 2^2 +2^4 +3 = 23

however, this then becomes a problem where any number can be written as a sum of 2s and 3s
7 = 2^1 + 2^1 + 3^1

unless you mean that we should be limited to representing numbers as only a 2^k term plus a 3^m term

Answer 20

,,as the sum of the exponential numbers of the numbers two and three "

how many times use the exponential number of 2 or/and 3 not specified

Answer 21

perfect, then this problem as been reduced to whether all numbers can be written as a sum of 2s and 3s

2k can be used to represent all of:
0 mod 6
2 mod 6
4 mod 6

3m can be used to represent all of:
3 mod 6

2 mod 6 + 3 mod 6 = 5 mod 6
and
4 mod 6 + 3 mod 6 = 1 mod 6

since every number in mod 10 can be rewritten in mod 6
it is possible to write every possible number as a sum of multiples of 2 mod 6 and 3 mod 6

Answer 22

we don't even need exponents if we are not limited by how many terms we can use

Answer 23

ok understand your idea - but a proof in this way ?

Answer 24

ah my apologies, I misunderstood something along the way: I thought you were asking whether every number can be written in such a way - which is true

primes are also included in this "every"
but primes specifically can also be written a 1 mod 6 and 5 mod 6
there is a proof of this somewhere, but I forget where I saw it

Answer 25

Here is a proof that says that any prime is a 1 mod 6 or 5 mod 6

Answer 26

then we can use what I stated before that 1 mod 6 can be a sum of 4 mod 6 and 3 mod 6
and 5 mod 6 can be a sum of 2 mod 6 and 3 mod 6

Answer 27

wwooaahhhwww - ty so much - this above posted - attached proof explain for me so much - and i think i can use it on my more idea about primes - @Angle can i collaborate with you in future in this subject - please !

Answer 28

I would be happy to collaborate in the future :)

primes are an interesting topic - I'm not too good at them, but I'm glad that the proof I found online was able to help you!

Answer 29

ok ty so much - you will see - i have a lot of ideas - new ideas - from the world of primes
but till today - till now i havent somebody with /to collaborate - talking about this - idk you was on OS too ? - so sorry i dont remember -

Answer 30

I was the moderator called jigglypuff314 when on OpenStudy
I just finished getting my bachelors in Mathematics and a masters degree in Education

I'm not too fond of primes, but my math classes taught me a lot. I had a great professor while at university who was also from Hungary. https://en.wikipedia.org/wiki/J%C3%B3zsef_Beck
He taught me a lot about combinatorics, which I am also not good at, but it was really fun.

Answer 31

ohhww really ?

Answer 32

yeah, I'm a middle school math teacher now in the US

Answer 33

do you see how little is the world ? you ve had a hungarian professor - nice very

Answer 34

indeed it is a small world! haha

Answer 35

what is interesting that i loved so much the combinatorics in my high school

Answer 36

then i remember right to solve a problem of combinatorics you needed 2-3 pages of A4

Answer 37

my favorite subject in high school was geometry
so representing the intersections of circles and lines was my favorite combinatorics topic

Answer 38

ok ty so much all your helps and ideas and you said i can tag write you in future when i like - i have posted questions in this subject of - about primes - yes ?

Answer 39

yes :) I do have a job during the day, but I will take a look at your prime questions when I can

Answer 40

ok ty so much - hnd - cya

Answer 41

IDK if I am reiterating the same thing that Angle and the OP discussed. But just to be clear I thought of posting here. Plus I'm not super familiar with modular arithmetic.

I assume that only positive exponents are allowed.

First we can prove that any even number \(\in\mathbb N\) can be represented in \(2n=\sum_{k=1}^{L}a_k2^k\) form where \(a_k\in\{0,1\}\) and some \(L\).
Then the rest is easy. Any odd number greater than or equal to 5 can be represented in the form \(2n+3\). If a prime number is greater than 2, then it is odd. So any prime number greater than 3 can be represented as sum of exponents of 2 and 3.