How To Prove That A Number Is Prime

Prime numbers, those enigmatic integers divisible only by 1 and themselves, have captivated mathematicians for centuries. Their seemingly random distribution and fundamental role in number theory make them both fascinating and crucial for various applications, from cryptography to computer science. Determining whether a given number is prime, however, isn't always straightforward, especially as numbers grow larger. This article delves into various methods for primality testing, explaining the underlying principles, advantages, and limitations of each approach. Whether you're a mathematics enthusiast, a student, or a programmer, understanding how to prove a number is prime will provide valuable insights into the world of number theory.

Introduction

Imagine you're given the number 101. Is it prime? A quick check reveals that it's not divisible by 2, 3, 5, or 7. But how far do you need to go? Do you have to check every number up to 100? This initial question highlights the core problem in primality testing: how can we efficiently determine if a number has any divisors other than 1 and itself? Fortunately, mathematicians have developed a range of techniques to address this challenge, each with its own strengths and weaknesses. From simple trial division to sophisticated probabilistic tests, the arsenal of primality tests offers a variety of approaches to tackle different scenarios.

This article will guide you through a journey exploring these methods, starting with basic approaches and progressing to more advanced techniques. We'll examine the logic behind each test, provide practical examples, and discuss their computational complexity. By the end, you'll have a comprehensive understanding of how to prove whether a number is prime and be able to choose the most appropriate method for a given situation.

Trial Division: The Simplest Approach

The most basic method for primality testing is trial division. The idea is straightforward: we try dividing the number n by every integer from 2 up to the square root of n. If we find any divisor, then n is composite (not prime). Otherwise, if we reach the square root of n without finding any divisors, then n is prime.

Steps for Trial Division:

1. Input: The number n we want to test for primality.
2. Optimization 1: Check if n is less than 2. If so, it's not prime (except for the case of 2 itself, which is prime).
3. Optimization 2: Check if n is even and greater than 2. If so, it's composite.
4. Iterate: Loop through integers i from 3 up to the square root of n, incrementing by 2 (we only need to check odd numbers after checking for divisibility by 2).
5. Check Divisibility: For each i, check if n is divisible by i (i.e., if n % i == 0).
6. Output: If any divisor is found, return "composite". If the loop completes without finding any divisors, return "prime".

Example: Let's test if n = 101 is prime using trial division.

• n is not less than 2.
• n is not even.
• We loop from i = 3 up to sqrt(101) ≈ 10.05, so we check 3, 5, 7, and 9.
• 101 % 3 = 2
• 101 % 5 = 1
• 101 % 7 = 3
• 101 % 9 = 2

Since none of these divisions result in a remainder of 0, 101 is prime.

Advantages of Trial Division:

• Simple: Easy to understand and implement.
• Effective for small numbers: Works well for relatively small values of n.

Disadvantages of Trial Division:

• Inefficient for large numbers: The time complexity is O(sqrt(n)), which becomes very slow as n increases. This makes it unsuitable for testing the primality of large numbers used in cryptography, for example.
• Doesn't provide factors: If n is composite, trial division only tells you that it's composite but doesn't necessarily provide all the factors.

Optimization with Prime Numbers:

We can optimize trial division further by only checking divisibility by prime numbers up to the square root of n. This is because any composite number can be expressed as a product of prime factors. If n has a divisor, it must have a prime divisor less than or equal to its square root.

Implementing Optimized Trial Division:

1. Generate a list of prime numbers up to the square root of n. This can be done using the Sieve of Eratosthenes (discussed later).
2. Instead of iterating through all odd numbers, iterate through the list of prime numbers.
3. The rest of the algorithm remains the same.

While this optimization improves the efficiency of trial division, its complexity still remains dependent on the size of n, especially considering the overhead of generating the prime number list.

The Sieve of Eratosthenes: Generating Prime Numbers

Before diving into more advanced primality tests, it's crucial to understand how to efficiently generate prime numbers. The Sieve of Eratosthenes is an ancient algorithm that provides a fast and elegant way to generate all prime numbers up to a given limit.

Algorithm for the Sieve of Eratosthenes:

1. Create a list: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n).
2. Mark the first prime: Let p be the first unmarked number in the list, which is 2.
3. Mark multiples: Mark all multiples of p greater than p itself in the list (2p, 3p, 4*p, ...). These are not prime.
4. Repeat: Find the next unmarked number in the list greater than p. If such a number exists, let it be the new p and repeat steps 3 and 4.
5. Output: When no unmarked numbers greater than p remain in the list, all the unmarked numbers remaining in the list are prime.

Example: Let's generate prime numbers up to 20 using the Sieve of Eratosthenes:

1. List: (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
2. p = 2
3. Mark multiples of 2: (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
4. p = 3
5. Mark multiples of 3: (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
6. p = 5
7. Mark multiples of 5: (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)

And so on, until all multiples of primes up to sqrt(20) are marked. The remaining unmarked numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

Advantages of the Sieve of Eratosthenes:

• Efficient for generating multiple primes: Particularly useful when you need a list of primes within a certain range.
• Simple to implement: Relatively easy to understand and code.

Disadvantages of the Sieve of Eratosthenes:

• Memory intensive: Requires storing a list of all numbers up to n, which can be a problem for very large values of n.
• Not suitable for testing single numbers: Primarily used for generating a list of primes, not for determining if a single number is prime.

Fermat's Little Theorem: A Probabilistic Test

Fermat's Little Theorem provides a different approach to primality testing. It states that if p is a prime number, then for any integer a not divisible by p, the following equation holds:

a^p-1 ≡ 1 (mod p)

This theorem can be used as a primality test. If we choose a random integer a and compute a^n-1 (mod n), and the result is not 1, then n is definitely composite. If the result is 1, then n is probably prime. It's important to note that Fermat's Little Theorem is a probabilistic test, meaning it can sometimes incorrectly identify composite numbers as prime (these are called pseudoprimes).

Steps for Fermat's Primality Test:

1. Input: The number n to test for primality and the number of tests to perform (different values of a).
2. Choose a base: Randomly choose an integer a such that 1 < a < n-1.
3. Compute a^n-1 (mod n): Calculate a^n-1 modulo n. This can be done efficiently using modular exponentiation.
4. Check the result: If a^n-1 (mod n) ≠ 1, then return "composite".
5. Repeat: Repeat steps 2-4 for a predetermined number of tests.
6. Output: If all tests pass (i.e., a^n-1 (mod n) = 1 for all chosen a), then return "probably prime".

Example: Let's test if n = 221 is prime using Fermat's Little Theorem with a = 3.

1. a = 3, n = 221
2. Calculate 3²²⁰ (mod 221). Using modular exponentiation, we find that 3²²⁰ (mod 221) = 1.

Since the result is 1, 221 might be prime. However, 221 = 13 * 17, so it's actually composite. This illustrates that Fermat's test can produce false positives.

Advantages of Fermat's Primality Test:

• Relatively fast: Significantly faster than trial division for large numbers.
• Simple to implement: Easy to understand and code.

Disadvantages of Fermat's Primality Test:

• Probabilistic: Can produce false positives (pseudoprimes).
• Carmichael numbers: There exist Carmichael numbers which are composite numbers that satisfy Fermat's Little Theorem for all a coprime to n. This means Fermat's test will always incorrectly identify Carmichael numbers as prime. The smallest Carmichael number is 561.

Miller-Rabin Primality Test: A More Robust Probabilistic Test

The Miller-Rabin primality test is a more sophisticated probabilistic test that addresses the limitations of Fermat's test. It's based on a stronger version of Fermat's Little Theorem and is less likely to produce false positives.

Underlying Principle:

If n is an odd prime, then n - 1 is even and can be written as 2^s * r, where r is odd. Then, for any integer a not divisible by n, either a^r ≡ 1 (mod n) or a^{2^jr} ≡ -1 (mod n) for some 0 ≤ j < s.

Steps for the Miller-Rabin Primality Test:

1. Input: The number n to test for primality and the number of tests to perform.
2. Check for small primes: If n is divisible by any small prime (e.g., 2, 3, 5, 7), return "composite" (except if n is equal to one of these small primes).
3. Find s and r: Write n - 1 as 2^s * r, where r is odd.
4. Choose a base: Randomly choose an integer a such that 1 < a < n-1.
5. Compute a^r (mod n): Calculate a^r modulo n.
6. Check the result:
   • If a^r ≡ 1 (mod n), then return "probably prime" for this test.
   • Otherwise, loop from j = 0 to s - 1:
     • Compute a^{2^jr} (mod n).
     • If a^{2^jr} ≡ -1 (mod n), then return "probably prime" for this test.
7. Output: If the loop completes without finding either a^r ≡ 1 (mod n) or a^{2^jr} ≡ -1 (mod n), then return "composite".
8. Repeat: Repeat steps 4-7 for a predetermined number of tests.
9. Final Output: If all tests pass, then return "probably prime".

Example: Let's test if n = 221 is prime using the Miller-Rabin test with a = 2.

1. n = 221, a = 2
2. 221 - 1 = 220 = 2² * 55, so s = 2 and r = 55.
3. Calculate 2⁵⁵ (mod 221). Using modular exponentiation, we find that 2⁵⁵ (mod 221) = 47.
4. Since 47 ≠ 1 (mod 221), we enter the loop:
   • j = 0: 2^{2⁰ * 55} (mod 221) = 2⁵⁵ (mod 221) = 47
   • j = 1: 2^{2¹ * 55} (mod 221) = 2¹¹⁰ (mod 221) = 174

Since neither 47 nor 174 is congruent to -1 (mod 221), we return "composite".

Advantages of the Miller-Rabin Primality Test:

• Highly accurate: The probability of incorrectly identifying a composite number as prime is very low, especially with multiple iterations.
• Widely used: Considered one of the most practical and efficient probabilistic primality tests.
• Less susceptible to pseudoprimes: Less likely to be fooled by pseudoprimes than Fermat's test.

Disadvantages of the Miller-Rabin Primality Test:

• Probabilistic: Still not a definitive primality test (though the probability of error is extremely low).
• More complex: More complex to implement than trial division or Fermat's test.

AKS Primality Test: A Deterministic Polynomial-Time Test

While probabilistic tests like Miller-Rabin are highly accurate and efficient, they don't provide a proof of primality. The AKS primality test, named after its inventors Agrawal, Kayal, and Saxena, is the first deterministic polynomial-time primality test. This means it guarantees to determine whether a number is prime or composite in a time that is polynomial in the number of digits of the input number, and it always gives the correct answer.

The AKS primality test is based on the following theorem:

An integer n > 1 is prime if and only if the polynomial congruence

(x - a)ⁿ ≡ (xⁿ - a) (mod n)

holds for all integers a coprime to n.

While the theoretical significance of the AKS test is enormous, its practical implementation is more complex. The original AKS algorithm had a relatively high time complexity, but subsequent improvements have made it more practical, though still generally slower than probabilistic tests for numbers of practical size.

Advantages of the AKS Primality Test:

• Deterministic: Provides a definitive proof of primality.
• Polynomial-time: Has a time complexity that is polynomial in the number of digits of the input number.
• Unconditional: Doesn't rely on any unproven hypotheses.

Disadvantages of the AKS Primality Test:

• Complex to implement: Significantly more difficult to implement than other primality tests.
• Slower than probabilistic tests for practical sizes: While polynomial-time, the constants involved make it slower than probabilistic tests for numbers commonly used in applications.

Elliptic Curve Primality Proving (ECPP): Proving Primality with Certificates

Elliptic Curve Primality Proving (ECPP) is a sophisticated method that not only determines whether a number is prime but also generates a certificate that can be used to independently verify the primality of the number. ECPP is based on the properties of elliptic curves over finite fields.

ECPP relies on Goldwasser-Kilian-Atkin (GKA) primality proving. It constructs an elliptic curve over the integers modulo n and searches for points on the curve that satisfy certain properties. If such points are found, it provides evidence that n is prime. The process is recursive, meaning that the primality of certain auxiliary numbers must also be proven. The resulting proofs can be independently verified, making ECPP a powerful tool for proving primality in a rigorous way.

Advantages of ECPP:

• Provides a certificate: Generates a certificate that can be used to independently verify the primality of the number.
• Rigorous proof: Provides a mathematical proof of primality.

Disadvantages of ECPP:

• Complex to implement: Requires a deep understanding of elliptic curves and finite fields.
• Computationally intensive: Can be computationally expensive, especially for large numbers.

Conclusion

Determining whether a number is prime is a fundamental problem in number theory with practical applications in cryptography and computer science. We've explored a range of methods, from the simple trial division to sophisticated algorithms like Miller-Rabin and AKS.

• Trial division is easy to understand but inefficient for large numbers.
• Fermat's Little Theorem provides a faster probabilistic test but is susceptible to pseudoprimes.
• The Miller-Rabin test is a more robust probabilistic test that is widely used in practice due to its high accuracy and reasonable speed.
• The AKS primality test is the first deterministic polynomial-time primality test, providing a guaranteed proof of primality, but its practical implementation is more complex and slower than probabilistic tests for typical number sizes.
• Elliptic Curve Primality Proving (ECPP) provides a certificate that can be used to independently verify primality, offering a rigorous approach but requiring significant computational resources.

The choice of the appropriate primality test depends on the specific application. For small numbers, trial division might suffice. For large numbers where speed is crucial, the Miller-Rabin test is often the best choice. When a rigorous proof of primality is required, AKS or ECPP can be used. Understanding the strengths and weaknesses of each method allows you to choose the most efficient and reliable approach for your needs.

How do you think the discovery of even more efficient primality tests will impact fields like cryptography in the future? And are you intrigued enough to explore the mathematics behind elliptic curves after learning about ECPP?