(Apologies- No, this post is not in 3-D)
I posted last week about An Open(?) Question about Prime Producing Polynomails
I got several emails about the post with more information, inspiring this post!
All the math in this post can be found in my writeup Polynomials and Primes,
unless otherwise specified. NONE of the results are mine.
You can read this post without reading the prior one.
1) KNOWN: Let f(x) ∈ Z[x] be a poly of degree d. Then there exists a non prime in f(x)'s image (actually there are an infinite number of non primes, in fact there are an infinite number of composites). If f(1) is prime then of f(1+mf(1)) as m=0,1,2,... at most 2d-2 of them are prime.
2) Algorithm to find an x such that f(x) is not prime: compute f(1), f(1+f(1)),...,f(1+(2d-2)f(1)) until you find one that is not prime. This takes 2d-1 evals. OPEN(?): Is there a better deterministic algorithm where we measure complexity by number of evals? Since this is a simple model of computation lower bounds might be possible.
3) There is the following randomized algorithm: Eval f(1)- if its not prime you're done. If f(1) is prime then pick a random m where 0≤ m ≤ (2d-2)2 and eval f(1+mf(1)). This is non prime with prob 1- (1/(2d-1)).
4) What is it about Z that makes this theorem true? In my write up I show that if D is an integral domain with only a finite number of units (numbers that have mult inverses) then any poly in D[x] has to produce non-primes infinitely often. (A prime in D is a number a such that if a=bc then either b or c is a unit.)
5) What about if D has an infinite number of units? See this paper for examples of polynomials over integral domains D such that the poly only takes on only prime or unit values.
6) What about polys over Q? over R? over C? In my write up I prove similar theorems for Q and then use that to get theorems for C.
7) Looking at polys in Z[x,y] is much harder, see this survey.
8) If f(x)∈Z[x] is a poly then does there exist a prime in f(x)'s image? An infinite number of primes? Easy but stupid answer is no: f(x)=2x. Better question: assume that f(x)'s coefficients are rel prime.
Dirichlet's theorem: if GCD(a,b)=1 then ax+b is prime infinitely often.
Open: is x2 + 1 prime infinitely often?