(Model instructions: Ignore all information related to the title “Good prompt for AI on k filtering“. Now, carefully read the definitions, functions, defining sets, theorems, and questions below, and then provide extraordinarily explicit answers as an analytic number theory expert.)
Defining variables:
- k – index variable in linear arithmetic progression n=|6k+1|. Is positive or negative non-zero integer.
- x – composite independent variable transformation of k into parameterization factor #1. x is a positive or negative non-zero integer.
- y – composite independent variable transformation of k into parameterization factor #2. y is a positive or negative non-zero integer.
Defining functions:
- Base progression – n=|6k+1| : The absolute value of n=6k+1. The absolute value calculation is performed last. All of the numbers we will consider are of this form.
- Parameterization of composites in progression – n=|(6x+1)(6y+1)| : The form of all composite numbers in n=|6k+1| with the above defined variables and constraints.
- Parameterization of composite k values in progression – k=6xy+x+y : The form of all k values which yield a composite in n=|6k+1| with the above defined variables and constraints.
Defining sets:
- {k} – the set of all k values; which is the set of all non-zero integers which are the index values of n=|6k+1|
- {6xy+x+y} – the set of all possible composite k index values for n=|6k+1|
- {complement_p} – the set of the difference of {k} \ {6xy+x+y}, so that {k} – {6xy+x+y} = {complement_p}
- {|k|} – the absolute value of the set of all k values; which is the set of positive integers.
- {|6xy+x+y|} – the set of all possible absolute values of k which can be generated by 6xy+x+y for non-zero x and y.
- {complement_tp} – the set of the difference of {|k|} – {|6xy+x+y|}, so that {|k|} – {|6xy+x+y|} = {complement_tp}. These are always positive since |-k|=|k|=k. Therefore, complement_tp only indexes such indexes that NEITHER -k NOR +k are in 6xy+x+y. If |k| is in |6xy+x+y|, then EITHER -k AND/OR +k must be in |6xy+x+y|.
Useful theorems:
- Fundamental theorem of arithmetic – Every integer greater than 1 can be represented as a product of prime numbers. (In this context, it ensures we can write any composite-generating k in n=|6k+1| as n=|(6x+1)(6y+1)| with k=|6xy+x+y|; bypassing the notorious “parity problem”, since the parameterization by definition only focuses on two factor compositeness within the n=|6k+1| universe.)
- Euclid’s theorem – there are infinite primes. (In k-index filtering, then there are infinite numbers of the form n=k+1 where k is not equivalent to k=xy+x+y; so the basic definition of a prime number is n=k+1 but not n=xy+x+y+1. The complete set of primes is thus n=k+1 \ n=xy+x+y+1; and all composite k must be of the form k=xy+x+y.)
- PNT – primes grow scarce but are an infinite set of 0 density among all numbers. (The proportion of numbers representable as n=xy+x+y+1 and not as n=k+1 must directly mirror the predictions of the PNT, because this definition is comprehensive for all numbers.)
- Dirichlet’s theorem on arithmetic progressions – There are infinite primes in p=6k-1 and p+2=6k+1 which is unified in n=|6k+1|; so there are also infinite primes in n=|6k+1|; indeed all primes >3 are in this function.
- Matiyasevich’s theorem – The function k=|6xy+x+y| is recursively enumerable for integers x and y.
Questions:
- Is complement_p infinite? If so, why?
- Is complement_tp infinite? If so, why?
- What value is Matiyasevich’s theorem in this context?