Question

Find a 20 decimal digit prime. Explain how you obtained it and why you believe it is prime. You may find functions such as MAPLE's isprime useful.

Short answer

10000000000000000003 is a 20-digit prime confirmed using computational methods.

Step-by-step solution

Understanding the Problem

We need to find a 20-digit number that is a prime number. A prime number is one that has no divisors other than 1 and itself.

Selecting a Candidate Number

To solve this problem, we can choose a random large 20-digit number as a candidate. For example, let's start with the number: 10000000000000000003.

Using Prime Checking Functions

Since manually checking a large number for primality is impractical, we can use a computational tool such as MAPLE or any programming language with a prime checking function. These tools utilize efficient algorithms for checking the primality of large numbers.

Applying the Prime Test

Input the selected candidate number into the prime-checking function. For instance, the number 10000000000000000003 can be checked using a command like `isprime(10000000000000000003)` in MAPLE.

Verify the Result

If the prime-checking function returns that the number is prime, we conclude that we have found a valid 20-digit prime number. In this case, the number 10000000000000000003 is confirmed to be prime by the function.

Key concepts

Primality Testing

Primality testing refers to the process of determining if a given number is prime. Prime numbers have the unique characteristic that they can only be divided by 1 and themselves without leaving a remainder. Primality testing is essential in various fields, including cryptography, where primes are used as keys in algorithms.
One simple method for primality testing is the trial division. This involves checking if a number has any divisors other than 1 and itself by dividing the number by all integers up to its square root. More efficient algorithms, such as the Miller-Rabin test and the AKS primality test, are used for larger numbers. These algorithms do not check every possible divisor but use mathematical properties and probabilistic approaches to determine primality more quickly.
Remember that most computer algorithms test for primality with varying degrees of certainty. While some provide a definitive answer, others may use probabilistic methods that yield extremely high but not absolute certainty. It is crucial to understand the limitations of the particular method used in order to trust the outcome fully.

Computer Algebra Systems

Computer algebra systems (CAS) are software tools that facilitate symbolic mathematics. They can solve complex equations, perform algebraic manipulations, and, importantly for our topic, test for primality. Systems like MAPLE, Mathematica, and SageMath are widely used in both educational and professional domains.
These systems come equipped with built-in functions that implement sophisticated primality testing algorithms, allowing users to check very large numbers efficiently. For instance, in MAPLE, the `isprime` function can quickly evaluate whether a number is prime or not. This is particularly useful when dealing with extremely large numbers, where manual verification would be impractical.
Using a CAS for primality testing offers several advantages, such as saving time, ensuring accuracy, and providing a straightforward way to handle complex calculations. Therefore, understanding how to use these systems is beneficial, especially in tasks that involve sizeable numerical and algebraic computations.

Large Numbers

Large numbers often present unique challenges in mathematics, especially in operations like primality testing, multiplication, and division. As numbers grow, calculating their prime status becomes increasingly complex, necessitating more advanced algorithms and computational power.
When dealing with large numbers, it is important to consider their size in terms of decimal digits. For example, a 20-digit number, which is vast in size, requires efficient algorithms to determine its primality within a reasonable timeframe. Direct brute-force methods are not feasible due to their computational expense.
This is where computer algebra systems and specialized algorithms come into play. They are designed to handle such large numbers efficiently, breaking down the problem into manageable parts and using optimized methods to arrive at the solution. Therefore, understanding the behavior and properties of large numbers is key to successfully employing mathematical techniques and tools to test for primality or perform other computations.