Integer partitions detect the primes

Area of Science:

• Number Theory
• Additive Number Theory
• Combinatorics

Background:

• Integer partitions are fundamental objects in additive number theory, representing a positive integer as a sum of positive integers.
• MacMahon's partition functions are well-studied combinatorial tools with applications in various mathematical fields.
• Identifying prime numbers is a central problem in number theory with significant implications for cryptography and computational mathematics.

Purpose of the Study:

• To investigate the relationship between integer partitions and the identification of prime numbers.
• To answer a question posed by Schneider regarding the use of partition functions in detecting primes.
• To establish novel prime-detecting equations using MacMahon's partition functions.

Main Methods:

• Utilizing the properties of integer partitions and MacMahon's partition functions.
• Developing and analyzing specific mathematical equations involving these partition functions.
• Proving the equivalence between the solutions of these equations and the primality of integers.

Main Results:

• Demonstrated that integer partitions can detect prime numbers through specific equations.
• Provided an example of a prime-detecting equation: an integer n ≥ 2 is prime if and only if [Formula: see text].
• Proved the existence of infinitely many prime-detecting equations with constant coefficients for MacMahonesque partition functions.

Conclusions:

• Established a novel and unexpected connection between integer partitions and prime number identification.
• The findings provide new tools for understanding and detecting prime numbers.
• The research opens avenues for further exploration in additive number theory and combinatorics.