University of Isfahan Mathematics and Society 2345-6493 10 4 2026 02 20 New results on Wolstenholmes theorem and its applications New results on Wolstenholmes theorem and its applications 105 131 29218 10.22108/msci.2025.141789.1669 FA Daniel Yaqubi Department of Computer Engineering, University of Torbat e Jam, , Iran Madjid Mirzavaziri Department of Mathematics, Ferdowsi University of Mashhad, Iran. Journal Article 2024 06 10 In 1862, Wolstenholme proved that for primes $p\geq 5$, $\binom{2p-1}{p-1}\equiv 1\pmod{p^3}$. This theorem is equivalent to the divisibility of the coefficients of the harmonic series \[1+\frac{1}{2}+\cdots+\frac{1}{p-1}\] by $p^2$. The far-reaching implications of Wolstenholme's theorem ignited the curiosity of mathematicians in the 19th century, leading to a surge of investigations into the divisibility properties of coefficients in rational fractions and binomial coefficients involving powers of primes. The emergence of Bernoulli numbers in this theorem and their intricate connection to binomial coefficients firmly established the roots of this mathematical domain in analytic number theory. In this paper, we embark on a journey to explore Wolstenholme's theorem for higher powers of primes and delve into the intricacies of these diverse proofs. The converse of Wolstenholme's theorem, first proposed by Jones, asserts that a natural number n satisfying the congruence $\binom{2n-1}{n-1}\equiv 1\pmod{p^3}$ must be prime. We conclude our exploration by examining the conditions of the converse of Wolstenholme's theorem and presenting several intriguing problems that beckon further investigation. In 1862, Wolstenholme proved that for primes $p\geq 5$, $\binom{2p-1}{p-1}\equiv 1\pmod{p^3}$. This theorem is equivalent to the divisibility of the coefficients of the harmonic series \[1+\frac{1}{2}+\cdots+\frac{1}{p-1}\] by $p^2$. The far-reaching implications of Wolstenholme's theorem ignited the curiosity of mathematicians in the 19th century, leading to a surge of investigations into the divisibility properties of coefficients in rational fractions and binomial coefficients involving powers of primes. The emergence of Bernoulli numbers in this theorem and their intricate connection to binomial coefficients firmly established the roots of this mathematical domain in analytic number theory. In this paper, we embark on a journey to explore Wolstenholme's theorem for higher powers of primes and delve into the intricacies of these diverse proofs. The converse of Wolstenholme's theorem, first proposed by Jones, asserts that a natural number n satisfying the congruence $\binom{2n-1}{n-1}\equiv 1\pmod{p^3}$ must be prime. We conclude our exploration by examining the conditions of the converse of Wolstenholme's theorem and presenting several intriguing problems that beckon further investigation. https://math-sci.ui.ac.ir/article_29218_9f068ee0c77ec2cb1b14e6c2295e2058.pdf