[Paper Review] On Kurepa's problems in number Theory

Aleksandar Ivić, Žarko Mijajlović|arXiv (Cornell University)|Dec 10, 2003

Analytic Number Theory Research5 references7 citations

TL;DR

This paper investigates Duro Kurepa's unsolved number theory problems, focusing on the left factorial hypothesis—that an odd prime p does not divide the sum of factorials from 0! to (p−1)!—and presents equivalent formulations in finite fields. It establishes connections to prime distribution, twin primes, and arithmetic progressions, while verifying the hypothesis computationally up to n = 1,000,000 and linking it to broader conjectures.

ABSTRACT

We discuss some problems in number theory posed by Djuro Kurepa (1907-1993), including his classical left factorial hypothesis that an odd prime $p$ does not divide $0! + 1! + ... + (p-1)!$.

Motivation & Objective

• To review and analyze lesser-known number theory problems posed by Duro Kurepa, particularly the left factorial hypothesis.
• To establish equivalent formulations of the left factorial hypothesis in finite fields GF(p), especially in terms of sums involving binomial coefficients and factorials.
• To investigate the sign behavior of sequences involving prime squares and neighboring primes, motivated by Kurepa’s second and third problems.
• To explore the infinitude of prime triplets and one-element sets in arithmetic progressions related to Kurepa’s problems.
• To connect Kurepa’s conjectures to major open problems such as the twin prime conjecture and the Bateman-Horn conjecture.

Proposed method

• Formulated the left factorial hypothesis as !n ≢ 0 (mod n) for all n > 2, and proved its equivalence to the condition that (p, !p) = 1 for all odd primes p.
• Derived equivalent statements in GF(p) using identities involving (−1)^k / k! and binomial coefficients, showing that the hypothesis holds iff certain sums are non-zero modulo p.
• Utilized computational verification by Slavić, Wagstaff, Mijajlović, and Gogić to confirm the hypothesis for all n < 1,000,000.
• Applied Bertrand’s postulate and the prime number theorem to analyze asymptotic behavior of prime sequences and derive inequalities involving p_n² and sums of earlier primes.
• Connected the sign of s_n = p_n² − p_{n−1} − p_{n+1} to the prime number theorem and known bounds on prime gaps.
• Linked the sign of π_n = p_n² − p_{n−1}p_{n+1} to the existence of good primes and twin primes, using Pomerance’s result on infinitely many good primes.

Experimental results

Research questions

• RQ1 Is it true that for every odd prime p, p does not divide the sum of factorials from 0! to (p−1)!?
• RQ2 Can the left factorial hypothesis be equivalently reformulated in terms of non-vanishing sums in the finite field GF(p)?
• RQ3 Are there infinitely many primes p for which p_n² > p_{n−1} + p_{n+1}?
• RQ4 Does the existence of infinitely many twin primes imply that π_n = p_n² − p_{n−1}p_{n+1} is negative for infinitely many n?
• RQ5 Are there infinitely many n ≡ 1 (mod 3) such that the set P(n) of primes x with x−2n, x, x+2n all prime is a singleton?

Key findings

• The left factorial hypothesis is equivalent to the condition that ∑_{k=0}^{p−1} (−1)^k (k+1)(k+2)⋯(p−1) ≢ 0 (mod p) for all odd primes p.
• The hypothesis has been computationally verified for all n < 1,000,000, with Gogić confirming it up to n = 10^6.
• The left factorial hypothesis is equivalent to the statement that ∑_{k=0}^{p−1} binom{p−1}{k} (k+1)⋯(p−1) ≢ 0 (mod p) for all primes p.
• The sequence s_n = p_n² − p_{n−1} − p_{n+1} is positive for all p_n ≥ 5, as shown using Bertrand’s postulate.
• The sequence π_n = p_n² − p_{n−1}p_{n+1} is negative for infinitely many n if the twin prime conjecture holds, due to the inequality p_n² − p_{n−1}p_{n+1} ≤ −2p_{n−1} + 4 < 0.
• The Bateman-Horn conjecture implies that there are infinitely many n ≡ 1 (mod 3) such that P(n) is a singleton, i.e., x−2n, x, x+2n are all prime for some x.