Prime Numbers, Coprimes & Fundamental Arithmeticअभाज्य संख्याएँ, सह-अभाज्य संख्याएँ एवं अंकगणित का मूलभूत प्रमेय
Properties of primes, twin primes, co-primes, prime factorizations, and primality testing algorithms up to square root N.
Overview: Prime Numbers, Coprimes & Fundamental Arithmetic अवलोकन: अभाज्य संख्याएँ, सह-अभाज्य संख्याएँ एवं अंकगणित का मूलभूत प्रमेय
• Prime Number: Natural number greater than 1 with exactly two distinct factors: 1 and itself.
• Smallest & Only Even Prime: 2.
• 1 is neither prime nor composite.
• Prime Distribution: Exactly 25 primes between 1 and 100; 15 primes between 1 and 50.
• 6k ± 1 Rule: All prime numbers greater than 3 can be expressed in the form 6k + 1 or 6k - 1 (converse is not necessarily true).
• Coprimes (Relatively Prime): Two numbers having an HCF of exactly 1. They need not be prime themselves (e.g., 8 and 9 are coprime).
• Primality Test: To test if N is prime, test divisibility by primes less than or equal to √N.
Every prime > 3 can be expressed as (6k +/- 1) | Check primality up to sqrt(N)