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

JohnK Wright

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%.

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%.
Paper Structure (14 sections, 1 theorem, 4 equations, 4 figures, 4 tables)

This paper contains 14 sections, 1 theorem, 4 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Wright-Euler Mersenne Exponent Hypothesis
  3. Derivation of the Wright-Euler Hypothesis
  4. Mathematical Analysis
  5. Connection to Classical Results
  6. Exact and Nearest-Integer Matches
  7. Comparison with Other Quadratic Polynomials
  8. Comparison to Exponential Model
  9. Early Approach with Integer $n$
  10. Graphical Analysis
  11. Proposed Mersenne Candidates
  12. Heuristic Algorithm for Candidate Selection
  13. Conclusion
  14. Appendix: Code and Resource Availability

Key Result

Theorem 1

For n >= 1, C(n) grows quadratically, with the bit length of 2^C(n) - 1 approximately C(n) + 1.

Figures (4)

  • Figure 1: Comparison of Wright-Euler matches (red squares), integer matches (orange triangles), actual exponents (blue dots), and exponential model (green line). MAE 614.0 for x = 30 to 52.
  • Figure 2: Preliminary predicted exponents for Mersenne primes #53--57 (early integer $n$ approach).
  • Figure 3: GIMPS discovery timeline, 52nd prime (p = 136279841).
  • Figure 4: Deviations d = |n - n_closest| for indices x = 30 to 35.

Theorems & Definitions (3)

  • Definition 1: Wright-Euler Mersenne Exponent Hypothesis
  • Theorem 1: Scale Properties
  • proof