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Integer partitions detect the primes.

William Craig1, Jan-Willem van Ittersum1, Ken Ono2

  • 1Department of Mathematics, United States Naval Academy, Annapolis, MD 21402.

Proceedings of the National Academy of Sciences of the United States of America
|September 20, 2024
PubMed
Summary
This summary is machine-generated.

Integer partitions unexpectedly identify prime numbers. Specific equations involving MacMahon

Keywords:
partitionsprimesquasimodular forms

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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.