8. Prime Numbers and GCD
The chapter discusses prime numbers, their properties, and a naive algorithm for primality testing. It also introduces the concept of the greatest common divisor (GCD) and details Euclid's GCD algorithm, highlighting its polynomial time complexity in relation to the number of bits required to represent integers. Key algorithms and their efficiencies are compared and explained.
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Sections
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What we have learnt
- Prime numbers are defined as integers greater than 1 that have no positive divisors other than 1 and themselves.
- The naive algorithm for primality testing operates in exponential time concerning the number of bits needed to represent a number.
- Euclid's GCD algorithm is an efficient method for computing the GCD of two integers, with polynomial time complexity.
Key Concepts
- -- Prime Number
- An integer greater than 1 that has no positive divisors other than 1 and itself.
- -- Composite Number
- An integer that has at least one positive divisor other than 1 and itself.
- -- Greatest Common Divisor (GCD)
- The greatest integer that divides two or more given integers without leaving a remainder.
- -- Euclid's Algorithm
- An algorithm for computing the GCD of two integers based on the principle that the GCD of two numbers also divides their difference.
Additional Learning Materials
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