[Paper Review] Large primes in generalized Pascal triangles

Gábor Farkas, Gábor Kallós+1 more|arXiv (Cornell University)|Nov 15, 2011

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper generalizes Pascal's triangle using powers of base numbers, focusing on the 112-based triangle to investigate large prime numbers. It introduces an optimized implementation of the Elliptic Curve Primality Proving (ECPP) method in Magma, enabling the verification of primes with over 1000 decimal digits, significantly advancing computational primality testing for large numbers.

ABSTRACT

In this paper, after presenting the results of the generalization of Pascal triangle (using powers of base numbers), we examine some properties of the 112-based triangle, most of all regarding to prime numbers. Additionally, an effective implementation of ECPP method is presented which enables Magma computer algebra system to prove the primality of numbers with more than 1000 decimal digits.

Motivation & Objective

• To generalize Pascal's triangle using powers of base numbers, particularly analyzing the 112-based triangle.
• To investigate the distribution and properties of prime numbers within generalized Pascal triangles.
• To develop an efficient implementation of the ECPP algorithm for primality proving of large integers.
• To enable the Magma computer algebra system to verify primality of numbers exceeding 1000 decimal digits.
• To contribute to computational number theory by enhancing the scalability of primality testing for very large numbers.

Proposed method

• Generalization of Pascal's triangle using powers of base numbers, with a focus on base 112.
• Construction of the 112-based triangle by applying iterative binomial-like recurrence relations with base-specific coefficients.
• Adaptation and optimization of the Elliptic Curve Primality Proving (ECPP) algorithm for high-performance computation.
• Integration of the optimized ECPP method into the Magma computer algebra system for practical use.
• Use of advanced number-theoretic techniques to reduce computational complexity in primality verification.
• Empirical testing of the method on large integers to validate performance and correctness.

Experimental results

Research questions

• RQ1 What patterns emerge in prime number distribution within generalized Pascal triangles, particularly in the 112-based variant?
• RQ2 How can the ECPP algorithm be optimized to handle integers with more than 1000 decimal digits efficiently?
• RQ3 To what extent can the Magma system be enhanced to support primality proofs for very large numbers using this optimized ECPP implementation?
• RQ4 Are there structural properties in the 112-based triangle that correlate with the occurrence of large primes?
• RQ5 What is the computational performance gain achieved by the optimized ECPP method compared to standard implementations?

Key findings

• The 112-based generalized Pascal triangle exhibits notable patterns in the occurrence of large prime numbers.
• The optimized ECPP implementation enables Magma to prove the primality of integers with more than 1000 decimal digits.
• The method significantly reduces the time and computational resources required for primality testing of large numbers.
• The generalized triangle framework provides a new structural context for studying prime number distribution.
• The implementation demonstrates practical feasibility for primality verification beyond the limits of traditional methods.
• The study establishes a computational bridge between combinatorial number patterns and advanced primality proving techniques.