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Loren Booda
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Do you know of an attempt to disprove the existence of a formula which predicts all prime numbers, as opposed to the accustomed attempt to derive one?
Can one show that an evaluation generating any prime in "one step" (such as including the determination of an unknown constant or knowledge of all previous primes) is impossible, and that all one-step, finite prime generating polynomials are limited in their range of producing primes? I'm asking whether it is possible to disprove that there exists a one-step mathematical generator of all primes?There exist a variety of formulas for either producing the nth prime as a function of n or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.
So other than finite polynomials, what functions do you consider non-iterative?Loren Booda said:Shmoe, thanks again for your help.
I suppose "one-step" can be defined as "non-iterative" (like the polynomials you just referred me to).
Hypothesis: given n, we cannot find every prime p(n) from anyone such non-iterative function.
And, of course, evaluating polynomials, computing exponents, multiplying numbers, and even adding numbers require some sort of iteration or related concept.Zurtex said:I mean functions like sin and cos all use iterations to calculate them numerically, along with almost all other interesting ones.
This hypothesis is false. I think you meant to say something like:Loren Booda said:Hypothesis: given n, we cannot find every prime p(n) from anyone such non-iterative function.
To quote Charlie Brown: "That's it!"We cannot find a non-iterative function that, when given n, computes the prime p(n).
Given x + y = zLoren Booda said:I'll give it a try:
Iterative functions fn, fn+1, fn+2 ... can be described by fn+1=g[fn] where g represents a function.
Loren Booda said:Let me attempt another approach.
Hypothesis: the most reduced function that generates all n primes, with n --> oo, has complexity approaching the most efficient process of factorization to ascertain those primes.
A prime counting function, also known as a prime number theorem or prime number formula, is a mathematical function that counts the number of prime numbers below a given value.
It is difficult to prove the nonexistence of a prime counting function because it requires showing that there is no possible formula or algorithm that can accurately count the number of prime numbers below a given value. This is a challenging task as there are infinite prime numbers and it is not known if there is a pattern or formula that can accurately predict their distribution.
One of the main pieces of evidence that suggests a prime counting function may not exist is the prime number theorem, which states that the number of prime numbers below a given value approaches infinity as the value increases. This suggests that there may not be a finite formula or algorithm that can accurately count the infinite number of prime numbers.
It is not currently known if the nonexistence of a prime counting function can be proven. Many mathematicians believe that it may not be possible to prove the nonexistence of a prime counting function, as it would require showing that there is no possible formula or algorithm that could accurately count all prime numbers.
If the nonexistence of a prime counting function could be proven, it would have significant implications for number theory and mathematics as a whole. It would mean that there is no finite formula or algorithm that can accurately predict the distribution of prime numbers, which would greatly impact our understanding of mathematics and the natural world.