We have three cases of primality by algebraic definition.
We will use these three cases to conceptualize prime number generation using algebraic functions with variables n, k, x, and y.
We will demonstrate that within these specific algebraic frameworks, the primality of n is entirely determined by whether its corresponding index k can be generated by a specific formula (xy, 2xy+x+y, or 6xy+x-y) representing composite numbers.
Case 1 – Fundamental definition of primes
- Our first definition is the basic definition of primality, so it covers all prime numbers greater than or equal to 2.
- n, k, x, y are positive integers ≥ 2.
- If n = 1k but n = xy ; then n is not prime.
- If n = 1k but n ≠ xy ; then n is prime.
- So, if k = xy, then n is not prime for a given n = 1k.
- But, if k ≠ xy, for a given n = 1k then n is prime.
Case 2 – Odd numbers
- Our second definition extends the case to odd numbers, so it covers all prime numbers greater than or equal to 3.
- n is a positive integer greater ≥ 3. k,x,y are all non-zero positive integers ≥1.
- If n = 2k+1 but n = 4xy+2x+2y+1, then n is not prime.
- If n = 2k+1 but n ≠ 4xy+2x+2y+1, then n is prime.
- So, if k = 2xy+x+y, then n is not prime for a given n = 2k+1.
- But, if k ≠ 2xy+x+y for a given n = 2k+1, then n is prime.
Case 3 – 6k±1 numbers
- Our third definition extends the case to numbers ±1 mod 6 (eg. 6k±1 numbers), so it covers all prime numbers greater than or equal to 5.
- n is a positive integer ≥ 5. k,x,y are all NON ZERO integers (may be negative).
- If n = |6k-1| but n = |36xy+6x-6y-1|, then n is not prime.
- If n = |6k-1| but n ≠ |36xy+6x-6y-1|, then n is prime.
- So, if k = 6xy+x-y, then n is not prime for a given n = |6k-1|.
- But, if k ≠ 6xy+x-y for a given n = |6k-1|, then n is prime.
(Explanation for case 3)
First, we demonstrate that for every n=6k-1, there is -n=6k+1 and vice versa.
- So, there is 5 in n=6k-1 for k=1, and there is -5 in n=6k+1 for k=-1.
- So, there is -7 in n=6k-1 for k=-1, and there is 7 in n=6k+1 for k=1.
- The sets are symmetrical, so the sets |{6k-1}|=|{6k+1}| have the same cardinality and absolute value which is reflected around 0.
- It is sufficient to use just the absolute value of 1 set to find all prime numbers in a symmetrical range. So we choose n = |6k-1| to classify all ±1 mod 6 numbers.
Next, we demonstrate there are 4 potential forms of composite emerging from (6x±1)(6y±1). We have:
- (6x-1)(6y+1) = 36xy+6x-6y-1 (always -1 mod 6 and produces numbers like 35)
- (6x+1)(6y-1) = 36yx+6y-6x-1 (always -1 mod 6 and produces the same values as the first equation, like 35, so let’s ignore it).
- (6x-1)(6y-1) = 36xy-6x-6y+1 (always 1 mod 6 and produces numbers like 25)
- (6x+1)(6y+1) = 36xy+6x+6y+1 (always 1 mod 6 and produces numbers like 49)
As we demonstrated before, for every n in 6k-1, there is -n in 6k+1, so this must also apply to the composites.
- (6(-1)-1)(6(1)+1) = -49 = |49|
- (6(1)-1)(6(-1)+(-1)) = -25 = |25|
So, n = |36xy+6x-6y-1| is sufficient to find all composites of 6k±1 by iterating through non-zero values of x and y.
So by reducing the equation and solving if, |6k-1|=|36xy+6x-6y-1|, then k = 6xy+x-y , and n cannot be prime.
In theory, you could create all the set of k=±1,±2,±3,±4…
Then, you can see if the sequential k value you created can be expressed as k = 6xy+x-y.
If it can, then n = |6k-1| is not prime.