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.
The relation
p + q = 2n²
can be equivalently rewritten as
p − n² = n² − q.
This shows that the prime pair (p, q) is symmetric around the perfect square n². Under this interpretation, the conjecture can be stated geometrically:
For every odd prime p, there exists a perfect square n² such that p has a prime mirror q at the same distance on the opposite side of n².
This square-centered formulation is mathematically equivalent to the original conjecture. It does not constitute an independent proof, but provides a geometric perspective that motivates the interpretation of perfect squares as symmetry centers in the distribution of primes.
| Primes Tested | Maximum p | Counterexamples | Max n |
|---|---|---|---|
| 1,000 | 7,919 | 0 | 90 |
| 10,000 | 104,743 | 0 | 279 |
| 100,000 | 1,299,721 | 0 | 915 |
| 1,000,000 | 15,485,863 | 0 | 2,925 |
The smallest n for each prime p is called the symmetry index s(p).
Observed relationship:
| Max p | √(Max p) | Max s(p) | Ratio s(p)/√p |
|---|---|---|---|
| 7,919 | 89 | 90 | 1.01 |
| 104,743 | 324 | 279 | 0.86 |
| 1,299,721 | 1,140 | 915 | 0.80 |
| 15,485,863 | 3,935 | 2,925 | 0.74 |
Conclusion: s(p) appears to grow as O(√p)
Among the first 1,000 primes:
| Condition | Count | Percentage |
|---|---|---|
| n divisible by 3 | 681 | 68.1% |
| n not divisible by 3 | 319 | 31.9% |
Expected by random chance: 33.3%
This suggests the formula 2n² - p favors producing primes when n ≡ 0 (mod 3).
If the minimal values of n were uniformly distributed modulo 3, one would expect approximately 33.3% of them to satisfy
n ≡ 0 (mod 3).
However, in the observed data, approximately 68.1% of the minimal n values fall into this class, representing a deviation of more than 2× the expected frequency. While no formal hypothesis test is performed here, the magnitude of this deviation strongly suggests a structural bias rather than random fluctuation.
This observation motivates further theoretical investigation into modular constraints induced by the quadratic form 2n².
Under this conjecture, every even number of the form
2n²
would be expressible as the sum of two odd primes. This constitutes a restricted instance of the classical Goldbach Conjecture, included here solely for contextual comparison.
No claim is made regarding the general validity of Goldbach’s Conjecture.
Exploratory visualizations and regression analyses related to square-centered symmetric prime pairs are provided in the analysis/ directory for transparency and reproducibility.
This directory contains exploratory scripts used to visualize and analyze square-centered symmetric prime pairs.
The scripts are provided for transparency and reproducibility.
They do not implement or propose a predictive or generative model.
The regression analyses are descriptive only and are intended to illustrate central tendencies under specific selection criteria.
python prime_symmetry_test.pyTo test with different number of primes:
python
from prime_symmetry_test import test_conjecture, analyze_divisibility
n_values, failed = test_conjecture(10000) # Test first 10,000 primes
analyze_divisibility(n_values)Can this conjecture be proven? Is s(p) < C√p for some constant C? Why do 68% of s(p) values divide by 3? Does a similar result hold for 2nᵏ where k > 2?
Research by Uğur Kandemiş (Vibe-X Protocol)