Predicting Mersenne Prime Exponents Using Euler's Quadratic Polynomial C(n) = n^2 + n + 41 with Nearest-Integer Rounding

TL;DR

This paper investigates the predictive power of Euler's quadratic polynomial C(n) = n^2 + n + 41, combined with nearest-integer rounding, for identifying candidate exponents of Mersenne primes, demonstrating promising success rates and potential for search space reduction.

Contribution

The study introduces a novel hypothesis linking Euler's polynomial to Mersenne prime exponents and evaluates its accuracy through empirical testing on known primes.

Findings

Seven exact matches of polynomial predictions with known exponents.

Four close approximations with small errors.

Significant reduction in search space for GIMPS testing.

Abstract

The Wright-Euler Mersenne Exponent Hypothesis proposes that Euler's quadratic polynomial C(n) = n^2 + n + 41, combined with nearest-integer rounding n_closest = round((-1 + sqrt(4p - 163))/2), identifies candidate exponents for Mersenne primes 2^p - 1. Applied to the 43 known Mersenne prime exponents with indices x = 10 through 52 (excluding p <= 31), the method produces seven exact matches (a 16.3% success rate, e.g., x = 38, p = 6972593 and x = 52, p = 136279841) and four close approximations (e.g., x = 34, p = 1257787, C(1121) = 1257803), with a mean absolute error of approximately 614 over the range x = 30 to 52. By comparison, an exponential regression model y = 11111.14 e^{0.1787x} captures the overall growth trend (R^2 approx 0.974) but yields no exact matches and a mean absolute error of 10,466,686. Graphical analysis, including scatter plots of C(n_closest) versus actual exponents and absolute deviations d = |n - n_closest|, demonstrates the hypothesis's precision when nearest-integer rounding is applied. From approximately 50 prime values of C(n) identified among 560 unique candidates, five cases with d < 0.1 are selected for targeted GIMPS testing, reducing the effective search space by approximately 74%.