A primality test for $Kp^n+1$ numbers and a generalization of Safe Primes and Sophie Germain Primes

Technical Summary: A Primality Test for $Kp^n + 1$ Numbers and a Generalization of Safe Primes and Sophie Germain Primes

Problem Statement
The paper addresses the computational challenge of determining the primality of integers of the form $N = Kp^n + 1$, where $p$ is a prime and $K < p^n$. While Fermat's primality test ($a^{N-1} \equiv 1 \pmod N$) is computationally inexpensive, it is probabilistic and insufficient for certifying primality on its own. Existing deterministic methods, such as Proth's theorem (for $N=K2^n+1$) and Pocklington's theorem, often require computing greatest common divisors (GCDs) or verifying complex conditions involving cyclotomic polynomials $\Phi_p$. The author seeks a method that simplifies these requirements, specifically aiming for a test that relies on a single modular exponentiation without GCD computations, and extends these concepts to define and analyze new families of primes analogous to Safe Primes and Sophie Germain Primes.

Methodology
The paper employs number-theoretic lemmas regarding the order of elements modulo $N$ and the divisibility properties of integers of the form $Ap+1$.

1. Generalization of Proth's Theorem: The author proves lemmas concerning the divisibility of $Ap+1$ by $Dp+1$ and the relationship between the order of an element modulo $N$ and its prime factors. These are used to derive new primality criteria.
2. Probabilistic and Heuristic Analysis: For the distribution of the newly defined prime families, the paper utilizes Hardy and Littlewood's Conjecture B and integral estimates to model the density of these primes. It compares theoretical asymptotic estimates against empirical counts of primes in specific intervals.
3. Computational Verification: The author performs extensive computations on primes with varying digit lengths (from small numbers up to 10,000 digits) to test the accuracy of the proposed distribution estimates.

Key Contributions and Results

• Simplified Primality Tests for $Kp^n + 1$:

  • Theorem 2.3: Provides a primality test for $N = Kp^n + 1$ (where $p^n \ge K$) requiring only two conditions: (i) $a^{Kp^n-1} \equiv L \not\equiv 1 \pmod N$ and (ii) $L^p \equiv 1 \pmod N$. This eliminates the need for the GCD computation required in previous generalizations (e.g., Grau et al., 2015).
  • Theorem 2.5: A further generalization allowing for a parameter $j$ ($0 \le j \le n-1$), which increases the number of valid bases $a$ that can prove primality.
  • Theorems 3.1 and 3.4: These theorems provide additional tests for specific ranges of $j$ and $n$ (e.g., $p^j \ge p^{n-j} \ge (N-1)^{1/3}$). These conditions allow for larger values of $j$, thereby increasing the probability that a randomly chosen base $a$ will satisfy the primality conditions, thus making the proof of primality more likely to succeed.
  • Corollaries: The paper identifies specific infinite families of integers (e.g., $N = 2Kp^2 + 1$ and $N = 2Kp^3 + 1$) where primality can be certified solely by verifying Fermat's condition $a^{N-1} \equiv 1 \pmod N$, provided specific bounds on $K$ and $p$ are met.

• Generalization of Safe Primes and Sophie Germain Primes:

  • Definition 4.1 ($a$-SafePrimes): A prime $N = 2Kp + 1$ is defined as an $a$-SafePrime if $2K < \log_a(2Kp+1)$. This ensures that the prime factor $p$ of $N-1$ is significantly larger than any other factor, mirroring the property of classical Safe Primes.
  • Definition of $a$-SophieGermainPrimes ($a$-SGP): A prime $p$ is an $a$-SGP if there exists at least one $K$ satisfying $2K < \log_a(2Kp+1)$ such that $2Kp+1$ is also prime.
  • Distribution Analysis: The paper derives asymptotic formulas ($\alpha_1, \alpha_2$) for the count of $a$-SafePrimes and estimates ($\alpha_3, \alpha_4, \alpha_5$) for the probability that a given prime is an $a$-SGP.
  • Conjectures on Distribution: Based on computations involving primes up to 10,000 digits, the author proposes:
    • Conjecture 5.1: The probability that a given prime $p$ is a 2-SophieGermainPrime converges to approximately 0.76 as $p \to \infty$.
    • Conjecture 5.2: A generalization of the above for $a$-SGPs, suggesting the probability converges to a limit dependent on $a$.

• Empirical Findings:

  • The paper presents tables comparing the actual counts of 2-SafePrimes and 2-SophieGermainPrimes against the proposed asymptotic estimates. The results show high accuracy even for very large numbers (100 to 10,000 digits).
  • The author applied these findings to Mersenne primes, identifying 21 out of the first 30 Mersenne primes that are also 2-SophieGermainPrimes, validating the utility of the estimates for searching large primes.
  • The probability that the largest known prime ($2^{82,589,933}-1$) is a 2-SophieGermainPrime is estimated to be approximately 0.7637.

Significance
The paper claims to provide a computationally simpler alternative to existing primality tests for the $Kp^n + 1$ family, specifically by removing the GCD step while maintaining rigor. Furthermore, it introduces a generalized framework for Safe and Sophie Germain primes, offering a statistical model for their distribution that appears robust even for extremely large integers. The author suggests these results are useful for generating large primes for cryptographic purposes and for understanding the distribution of primes with specific structural properties. The work relies on established number theory (Proth, Pocklington, Hardy-Littlewood) and extends these through new lemmas and empirical validation.