This conjecture is not proposed as an arbitrary numerical pattern, but is motivated by heuristic considerations arising from the interaction between prime distribution and quadratic growth.
The quantity 2n² defines a symmetric quadratic center on the number line. Unlike linear centers (as in classical Goldbach-type problems), quadratic centers grow fast enough to provide an expanding search space for admissible prime pairs, while remaining sparse enough to avoid trivial saturation. This balance makes 2n² a natural candidate for studying structured prime symmetry.
From a probabilistic number-theoretic perspective, the Bateman–Horn heuristic suggests that polynomial expressions of the form
q(n) = 2n² − p
should attain prime values with asymptotic density proportional to
~ 1 / log(2n²),
provided that no local congruence obstructions systematically eliminate candidates. For odd primes p, the expression 2n² − p avoids trivial modular exclusions, making the existence of admissible n values heuristically plausible.
Empirically, the fact that relatively small values of n suffice across large datasets suggests that the expected waiting time for a prime hit grows slowly compared to the quadratic scale.
This repository investigates the following conjecture:
Conjecture.
For every odd prime ( p \ge 3 ), there exists an integer ( n \ge 2 ) such that
[ q = 2n^2 - p ] is also prime.
Equivalently: [ p + q = 2n^2 ] so the primes (p) and (q) are symmetric around the square center (n^2).
- The square (n^2) acts as a symmetry center.
- Prime pairs ((p,q)) appear as equal-distance deviations: [ p = n^2 - d,\quad q = n^2 + d ]
- The center (n^2) itself cannot be prime (for (n>1)), so primality is realized only through symmetric offsets.
This makes (n^2) a binding but non-collapsing constraint:
it enforces balance without destroying primality.
The main script:
- Generates the first (N) odd primes.
- For each prime (p), searches for an integer (n) such that
(q = 2n^2 - p) is prime. - By default, computes the true minimal symmetry index [ s(p) = \min{n \ge 2 : 2n^2 - p \text{ is prime}}. ]
An optional heuristic mode is provided for faster scanning, but it does not guarantee minimality and is clearly labeled as such.
-
minimal (default)
Scans (n = 2,3,4,\dots) in increasing order.
Guarantees that the returned (n) is the smallest possible. -
fast (heuristic)
Prioritizes (n \equiv 0 \pmod{3}) to speed up searches.
Useful for large experiments, but not guaranteed to return the minimal (n).
The conjecture itself concerns existence, not minimality;
minimality is used here as a canonical, reproducible choice.
We also test whether the polynomial [ q(n,p) = 2n^2 - p ] is locally obstructed modulo small integers.
A complete scan for all moduli ( m \le 31 ) shows:
- No nontrivial local obstructions.
- The only obstruction occurs for (p \equiv 0 \pmod{2}), which is outside the scope of the conjecture (odd primes only).
Thus, the conjecture is locally consistent: it does not collapse modulo small integers.
Empirically, many minimal symmetry indices satisfy ( n \equiv 0 \pmod{3} ).
This is not a local obstruction. Instead, it reflects a modular narrowing: certain residue classes of (n) avoid forced divisibility by 3 in (q = 2n^2 - p).
In other words:
- mod 3 creates a preferred escape route,
- not a barrier.
- No counterexample found for the tested ranges.
- No small-modulus local obstruction detected.
- The conjecture remains open.
This repository provides experimental evidence and geometric intuition, not a proof.
Run the main script to test the conjecture on the first (N) odd primes and analyze symmetry indices.
Research by Uğur Kandemiş (Vibe-X Protocol)