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

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%.