Alsings hypothesis of integers bigger than 2

[HAP]

I have a found a hypothesis which I would like you to look at, and perhabs (dis)prove..

-----------------
All integers (n) bigger than 2 (3, 4, 5, 6, ...) be descriped as:

n = (p_1 * p_2 * ...) + k

where all p and k are primes, but also include 1. Notice that k < (p_1 * p_2 * ...), and that no prime may occur more than once.

-----------------
Here is what I found out until now:

1) All values of the equation can be decriped as 2x added an optional value of 1:

All of the primes (p_i and k) bigger than 2, can be descriped with the equation aswell. I you repeat doing this you will end up with an equation consisted of 2's and 1's added and multiplied together. Check the example:

10 = 7+3
3 = 2+1
7 = 2*3+1 = 2*(2+1)+1
<=>
10 = (2*(2+1)+1)+(2+1) = (2*2+2*1)+1+2+1 = (2*3)+2+(1+1) = 2*4+2 = 2*5

This can be done with all the equations of n, but sometimes 2x is added the value of 1..

2) None of n's prime factors occur in the equation for n:

Let p_n be a prime factor to n. If p_n occures in (p_1 * p_2 * ...) the result would also be divisible by p_n. This means that no k-value exists (obeying the requirements), which would make n divisible by p_n (k cannot be p_n or divisible to p_n). Since this is a contradiction in terms, p_n cannot occur in (p_1 * p_2 * ...).
Now try to apply the value of p_n to k. Then (p_1 * p_2 * ...) cannot be p_n or divisible to p_n (since the prime cannot occur twice). This gives a value for n, which is not divisible by p_n and results in a contradiction in terms, onces again.

-----------------
I have showed that all values of the equation can be broken down to 2x added an optional value of 1. But to prove this hypothesis you have to show that all values of 2x added an optional value of 1, also can be descriped with my equation!

Enjoy..

 

[HAP]

By the way.. I am now finished running a homemade program made to test if this hypothesis is true for all numbers up to 500; and indeed it is!

 

[M98Ranger]

Thank you for sharing your hypothesis with me. It is an interesting concept to consider that all integers bigger than 2 can be described as a combination of primes and 1. However, I have a few concerns about this hypothesis that I would like to discuss.

Firstly, I am not sure if your statement that "all values of the equation can be broken down to 2x added an optional value of 1" is completely accurate. While it may be true for some values, I believe there are also cases where this may not hold. For example, the number 8 can be written as 2x(2+1) but it can also be written as 2x(3+1). This means that there may be other ways to represent integers bigger than 2 that do not fit into your equation.

Additionally, I am not convinced by your second point that none of n's prime factors occur in the equation for n. While it may be true for some cases, I do not believe it is true for all cases. For example, let's take the number 15. This can be written as (2x2x2)+(3+1) which follows your equation. But it can also be written as (2x2x2)+(2x2+1) which includes the prime factor of 2 twice. This means that your hypothesis may not hold for all integers bigger than 2.

In order to prove your hypothesis, you would need to show that all integers bigger than 2 can be represented in the form of (p_1 * p_2 * ...) + k where all p and k are primes, including 1, and that no prime may occur more than once. This would require a comprehensive mathematical proof that takes into account all possible cases and values of n.

Overall, I think your hypothesis is an interesting idea to explore, but it would require further investigation and proof to be considered a valid mathematical concept. I encourage you to continue exploring and refining your hypothesis to see if it holds true for all integers bigger than 2.