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Using a parity-sensitive sieve to count prime values of a polynomial
1Scarborough College, University of Toronto, Scarborough, ON, M1C 1A4, Canada.
Summary
Researchers developed new methods to prove that certain irreducible polynomials generate infinitely many prime numbers. This breakthrough addresses the challenge of finding primes in "thin" integer sequences, like x(2) + y(4).
Area of Science:
- Number Theory
- Analytic Number Theory
- Algebraic Geometry
Background:
- Irreducible polynomials with integer coefficients are conjectured to produce infinitely many prime values.
- Proving this conjecture is challenging, especially for
Purpose of the Study:
- To develop new methods for proving the infinitude of primes in specific thin integer sequences.
- To rigorously demonstrate the asymptotic formula for prime values of the polynomial x(2) + y(4).
Main Methods:
- A novel sieve method that overcomes the parity problem in sieve theory.
- Harmonic analysis tailored to the properties of biquadratic polynomial sequences.
Main Results:
- The study successfully demonstrates the asymptotic formula for prime values of the polynomial x(2) + y(4).
- The developed methods are applicable to an infinite class of similar polynomials.
Conclusions:
- New techniques provide a rigorous proof for prime generation in a previously intractable class of polynomials.
- This work advances the understanding of prime distribution within thin integer sequences.