This question is inspired by the classical behavior of Euler’s polynomial $$ \mathbf{f(x) = x^2 - x + 41}, $$ which is well-known for producing prime values for integer inputs $x = 0$ to $39$, and is linked to the class number one field $\mathbf{\mathbb{Q}(\sqrt{-163})}$. It is one of several quadratics of the form $x^2 + x + a$ or $x^2 - x + a$ associated with imaginary quadratic fields of class number one.
After studying the sequence generated by $f(x)$, I observed a phenomenon I refer to as prime inheritance:
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Previous Conjecture (Prime Inheritance Property) from previous question (maybe related)
For $\mathbf{f(x) = x^2 - x + 41}$, every composite output of $f(n)$ has at least one prime factor that appeared as an earlier prime output of the same sequence.
This seems to hold all the way up to $x = 10^5$. I call this prime inheritance because composite outputs appear to be “composed of” primes that the sequence already emitted.
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Observation (possible new conjecture) Prime-Generating Subsequences
Whenever a new prime factor appears — one not previously observed in earlier outputs of distinct subsequences (i.e., those that are not repeats of the main sequence) — it becomes the beginning of a new prime-generating subsequence of at least 3 consecutive primes) in the first 3 outputs.
Example: • The prime $163$ first appears at $x = 81$ as a factor of a composite value. • Then it begins the sequence: $$ \mathbf{163,\ 167,\ 179,\ 199,\ \dots}, $$ which are primes produced by the formula: $$ \mathbf{f_1(x) = 4x^2 + 163}. $$
Similar branches include: • $\mathbf{f_2(x) = 9x^2 - 3x + 367}$ • $\mathbf{f_3(x) = 9x^2 + 21x + 379}$ • $\mathbf{f_4(x) = 4x^2 + 4x + 653}$ • etc.
Each of these begins when a new prime appears in the factorization of $f(n)$, and then often goes on to produce a sequence of prime outputs at consecutive or near-consecutive indices.
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Further Phenomenon: Subsequence Propagation
We observe the following: • The main function $\mathbf{f(x) = x^2 - x + 41}$ not only generates primes directly but its composite values contain prime factors that recursively seed new subsequences. • These subsequences themselves seem to follow structured polynomial forms. • Possible new conjecture. the first few outputs of each of these new polynomials are again prime, creating prime trees rooted in the main sequence.
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Related Question:
Let $f(x) = x^2 - x + 41$, and define its set of prime outputs as $\mathbf{P}$. Let $\mathbf{C \subset \mathbb{Z}}$ be the set of $x$ for which $f(x)$ is composite. Is it true that:
For all $x \in C$, every prime factor of $f(x)$ appears as a prime output at some smaller $n < x$ in either: • The main sequence $f(n)$ • Or one of its inheriting subsequences that are not just repeats of $f(x)$?
And more generally:
Does this phenomenon of prime inheritance and sub-sequence formation occur in all class number one quadratic polynomials?
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Summary of What Was Done: • Computed values of $\mathbf{f(x) = x^2 - x + 41}$ up to $x = 10^5$ • Verified the inheritance conjecture computationally • Identified the emergence of new prime-generating sequences from new prime factors • Observed similar behavior for other class number one polynomials such as $\mathbf{x^2 + x + 17}$
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What I’m Looking For: • Can this conjecture be proven or refuted? • Is this inheritance tree structure known in the literature? • Is there an algebraic number theory framework (e.g. Hilbert class fields or splitting primes) that predicts this behavior? • Is this behavior unique to class number one, or does it appear more broadly?
Any insights, references, or counterexamples would be greatly appreciated.
While many of these subsequences begin with long stretches of prime outputs, we do not claim they are infinite. In fact, we expect most of them to eventually produce composite values. What’s striking is how they appear to be seeded by previously unseen prime factors, suggesting a structured method of prime propagation — even if the prime streaks are finite.
Edit: I initially believed this question to be distinct from my earlier one because if this conjecture holds, it would imply that there exist infinitely many formulas (or subsequences) whose first consecutive outputs are prime.
That said, I now understand that this may be covered (at least in principle) by the reasoning in the answer to my previous question. I will study that answer carefully to better understand the full implications.
I sincerely apologize if this post caused duplication or wasted anyone’s time. I appreciate all feedback and thank you for your attention and patience.
