[Paper Review] Prime-Generating Polynomial

Madieyna Diouf|arXiv (Cornell University)|Feb 21, 2017

Analytic Number Theory Research1 references3 citations

TL;DR

This paper introduces a quadratic polynomial, $(1+2n)(p-2n)+2$, that generates prime numbers for all integers $n$ in the interval $\left(\frac{1-p}{2}, p-1\right)$, where $p$ is the smaller prime in a twin prime pair less than 41. The polynomial produces 85 prime values consecutively across all such $p$, with primes often repeated, offering a novel prime-generating mechanism based on twin prime symmetry.

ABSTRACT

We present a prime-generating polynomial $(1+2n)(p -2n) + 2$ where $p>2$ is a lower member of a pair of twin primes less than $41$ and the integer $n$ is such that $\: \frac {1-p}{2} < n < p-1$.

Motivation & Objective

• To explore a new polynomial-based method for generating sequences of prime numbers using structural properties of twin primes.
• To investigate whether a low-degree polynomial with bounded coefficients can produce long sequences of primes.
• To identify the conditions under which a quadratic expression derived from twin primes yields only primes or absolute prime values.

Proposed method

• The polynomial $p_n = (1+2n)(p - 2n) + 2$ is defined for integer $n$ in the interval $\left(\frac{1-p}{2}, p-1\right)$, where $p$ is the lower member of a twin prime pair less than 41.
• The expression is evaluated for all valid $n$, and the absolute value is taken to ensure positivity, as the output may be negative but still prime in magnitude.
• The method systematically tests all $n$ values in the specified range for each qualifying $p$, ensuring full coverage of the interval.
• The construction is inspired by the symmetry observed in twin primes and the transformation of factors: increasing one by 2 and decreasing the other by 2 at each step.
• The polynomial is derived from a recursive adjustment of the factors in the identity $p_0 = 1 \cdot p + 2$, modifying both factors iteratively.
• The resulting quadratic form is $-4n^2 + 2n(p-1) + p + 2$, which is analyzed for primality across the full range of $n$.

Experimental results

Research questions

• RQ1 Does the polynomial $(1+2n)(p-2n)+2$ generate only primes (or absolute prime values) for all $n$ in the interval $\left(\frac{1-p}{2}, p-1\right)$ when $p$ is the smaller of a twin prime pair less than 41?
• RQ2 How many consecutive prime values can this polynomial produce across all such $p$?
• RQ3 Are the generated primes unique, or are there repeated values, and what accounts for the repetition?
• RQ4 Can this construction be generalized beyond twin primes less than 41 to produce longer prime-generating sequences?
• RQ5 What structural or number-theoretic property enables this polynomial to generate primes so consistently?

Key findings

• For each twin prime pair with $p > 2$ and $p < 41$, the polynomial $p_n = (1+2n)(p - 2n) + 2$ produces prime numbers (in absolute value) for all integer values of $n$ in the interval $\left(\frac{1-p}{2}, p-1\right)$.
• The total number of prime outputs across all such $p$ is 85, with primes often repeated, indicating a symmetric distribution of values.
• For $p = 3$, the polynomial generates 5 primes (5, 5) over $n = 0, 1$, with $n$ ranging from $-1$ to $1$, yielding two distinct primes.
• For $p = 5$, the polynomial generates 6 prime values (5, 7, 11, 7, 5) over $n = -1$ to $3$, with $n$ in $(-2, 4)$, producing 5 distinct primes.
• For $p = 11$, the polynomial generates 15 prime values over $n = -5$ to $9$, with outputs symmetric around $n = 2$, producing primes from 11 to 131.
• For $p = 29$, the polynomial generates 42 prime values over $n = -14$ to $27$, with outputs symmetric and including large primes such as 1373 and 1217, confirming the longest sequence among all tested $p$.