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This study proves two supercongruences related to Domb numbers. The findings advance number theory by confirming conjectures on truncated sums involving these mathematical objects.

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Area of Science:

  • Number Theory
  • Combinatorics
  • Algebraic Number Theory

Background:

  • Supercongruences are a class of congruences that hold in a stronger sense than usual.
  • Domb numbers and their associated truncated sums are objects of interest in number theory and combinatorics.
  • Zhi-Hong Sun recently proposed several conjectures regarding these sums.

Purpose of the Study:

  • To rigorously prove two specific supercongruence conjectures posed by Zhi-Hong Sun.
  • To establish new results concerning truncated sums involving Domb numbers modulo prime powers.

Main Methods:

  • Utilizing congruences involving specialized Bernoulli polynomials.
  • Applying properties of harmonic numbers and binomial coefficients.
  • Employing techniques from hypergeometric summation and transformation identities.

Main Results:

  • Successfully proved two conjectured supercongruences for truncated Domb number sums.
  • Demonstrated the validity of these congruences modulo primes p > 3.

Conclusions:

  • The paper confirms significant conjectures in the area of number theory.
  • The methods provide a framework for analyzing similar congruences in the future.