001350 What is a Prime Number?





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Before we address "What’s a Prime Number?”, we start by exploring the concept of numbers. Numbers can be classified according to various criteria. We list these criteria in alphabetically ascending order.



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Arabic Numbers vs. Roman Numbers


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Carmichael Numbers and Non-Carmichael Numbers


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Complex Numbers vs. Non-Complex Numbers


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Composite Numbers vs. Non-Composite Numbers (Prime Numbers)


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Even Numbers vs. Odd Numbers


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Graham Numbers and Non-Graham Numbers


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Integer Numbers vs. Non-Integer Numbers (Numbers with Fractional Part)


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Natural Numbers vs. Non-Natural Numbers


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One-Dimensional Set Numbers vs. Multi-Dimensional Set Numbers


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Perfect Numbers and Non-Perfect Numbers


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Perfect Numbers vs. Non-Perfect Numbers


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Positive Numbers vs. Negative Numbers


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Positive Numbers vs. Non-Positive Numbers (Negative Numbers)


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Radical Numbers vs. Non-Radical Numbers


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Rational Numbers vs. Non-Rational Numbers (Irrational Numbers)


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Real Numbers vs. Non-Real Numbers (Imaginary Numbers)


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Sequential Numbers vs. Non-Sequential Numbers (Random Numbers)


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Set Numbers vs. Non-Set Numbers


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Special Numbers and Non-Special Numbers (PI; e; etc.)


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Special Numbers vs. Non-Special Numbers


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Ten-based Numbers (Decimal) vs. Non-Decimal Numbers (Binary, Octal, Duodecimal, Hexadecimal)


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Terminating Numbers (Numbers with Finite Lengths / Numbers with Defined Lengths / Ending Numbers) vs. Non-Terminating Numbers


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Terminating Numbers vs. Non-Terminating Numbers


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Transcendental Numbers and Non-Transcendental Numbers


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Transcendental Numbers vs. Non-Transcendental Numbers


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Whole Numbers vs. Non-Whole Numbers



A number can belong to one or more of these categories. Let’s begin by defining Natural Numbers. There are multiple ways to define Natural Numbers. The following is one definition.



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A natural number is a number that matches the following three specifications:

a.   A real number.

b.   Greater than zero.

c.   Has no fractional part.


Examples of natural numbers include:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and so on.

Essentially, a Natural Number is a Positive Integer.

For instance, when counting people in a classroom or a movie theater, we are using natural numbers.



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What Is a Prime Number?


There are multiple ways to define Prime Numbers. The following is one definition.


A Prime Number is a number that meets the following four specifications:


a.   A Natural Number (Positive Integer).

b.   Greater Than 1.

c.   An Odd Number (Except for 2).

d.   When divided by any other Natural Number and less than itself and greater than 1, does not result in a Natural Number.


Most of the Natural Numbers are NOT Prime Numbers. Thus We apply Specification 1 to the tested number. If the tested number does not meet Specification 1, then the tested number is NOT a Prime Number. We skip Specification 2, Specification 3, and Specification 4.


We apply Specification 2 to the tested number. If the tested number does not meet Specification 2, then the tested number is NOT a Prime Number. We skip Specification 3 and Specification 4.


We apply Specification 3 to the tested number. If the tested number does not meet Specification 3, then the tested number is NOT a Prime Number. We skip Specification 4.


If the tested number meets Specification 1, Specification 2, Specification 3, and Specification 4, then the tested number is a Prime Number.


We test number “2”:

a.   “2” is a natural number.

b.   “2” is greater than 1.

c.   “2” is NOT an odd number but the exception applies.

Therefore “2” is a Prime Number.


We test number “3”:


a.

“3” is natural number.


b.

“3” is greater than 1.


c.

“3” is an odd number.


d.1.

3 / 2 = 1.5

“1.5” is NOT a Natural Number.

Thus “3” is a Prime Number.



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Composite Numbers


A Composite Number is a number that meets the following three specifications:


a.

A Natural Number (Positive Integer).


b.

Greater Than 2.


c.

When divided by any other natural number, and less than itself, and greater than 2, does result in one or more natural number.



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Historical Development of Prime Numbers

Prime numbers have intrigued mathematicians for centuries. The study of prime numbers dates back to ancient times, with contributions from many different cultures, including the Greeks, Chinese, and Indians. One of the earliest contributions to the theory of prime numbers comes from Euclid, who lived around 300 BCE. Euclid proved that there are infinitely many prime numbers, a groundbreaking result in ancient mathematics.


In Chinese mathematics, dating back to around 500 BCE, a notable assertion was made that if a number "n" divides the expression "2 raised to the power of n minus one, minus one," then "n" must be a prime number. While this assertion holds for all numbers less than and including 340, it was eventually proven false in the year 1820. The number 341 divides the expression "2 raised to the power of 340 minus one," yet 341 is not a prime number.


Despite this failure, the assertion that if "n" is a prime number, then "n" divides "2 raised to the power of n minus one, minus one" is true. This idea was later generalized in Fermat's Little Theorem, which was introduced by Pierre de Fermat in 1640. Fermat's theorem states that for any prime number "p" and any integer "a," the expression "a raised to the power of p minus one" is congruent to one modulo p.


Fermat Numbers and Their Disproval

Pierre de Fermat, a prominent French lawyer and mathematician, developed a formula to generate prime numbers, known as Fermat numbers. Fermat's formula is expressed as:


Fn = two raised to the power of two raised to the power of n, plus one.


Fermat believed that this formula would always yield prime numbers. However, in 1732, Leonhard Euler proved that Fermat's assumption was incorrect. Euler demonstrated that the fifth Fermat number (F5) was not a prime number. F5 equals 4,294,967,297, which can be factored into 641 multiplied by 6,700,417. This discovery shattered Fermat's belief that his formula always generated prime numbers.


Nevertheless, Fermat's contributions to number theory, including Fermat's Last Theorem and Fermat's Little Theorem, left a lasting impact on mathematics. His work laid the groundwork for many future discoveries in the field.


Euler's Theorem and Generalization

Leonhard Euler, a Swiss mathematician, made significant advancements in number theory. After disproving Fermat's formula in 1732, Euler went on to generalize Fermat's Little Theorem in 1736. Euler's generalization is known as Euler's Theorem, which states that if two numbers "a" and "n" are relatively prime, meaning their greatest common divisor is one, then the expression "a raised to the power of Euler's totient function of n" is congruent to one modulo n.


Euler's Theorem has profound implications in number theory and is widely used in fields such as cryptography and computer science.


Discovery of Carmichael Numbers

In 1910, mathematician R. D. Carmichael discovered a new class of numbers known as Carmichael numbers. These numbers are also called absolute pseudoprimes because they exhibit certain properties that make them appear to be prime numbers under specific conditions, even though they are composite. Carmichael numbers play an important role in number theory and the study of pseudoprimes.


Mersenne Primes and Marin Mersenne

Another famous class of prime numbers is Mersenne primes, named after the French monk and mathematician Marin Mersenne. A Mersenne prime is a prime number that can be written in the form:


Mp = two raised to the power of p, minus one, where p is itself a prime number.


For example, M31, where p equals 31, is a Mersenne prime, and its value is 2,147,483,647. The largest prime number to be discovered without the aid of electronic computers was M127, found in 1876. Since then, larger Mersenne primes have been discovered using powerful computers, and the search for new primes continues.


Bernhard Riemann and the Distribution of Prime Numbers

In 1859, Bernhard Riemann, a German mathematician, presented a paper to the Berlin Academy titled "On the Number of Prime Numbers Less Than a Given Quantity." This work laid the foundation for the Riemann Hypothesis, one of the most important unsolved problems in mathematics. Riemann's hypothesis suggests that the distribution of prime numbers follows a particular pattern, and proving this hypothesis could unlock new insights into the nature of prime numbers.


Largest Known Prime Numbers

With advances in computing, the discovery of prime numbers has accelerated. In 1999, the largest known prime number was a Mersenne prime, M3021377, which equals "two raised to the power of 3,021,377 minus one." This prime number was discovered by Ronald Clarkson, a 19-year-old student at California State University.


By the year 2000, it was confirmed that all prime numbers under "ten raised to the power of 20" were known to humans. Today, larger prime numbers are continually being discovered through distributed computing efforts and advanced algorithms.


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Applications of Prime Numbers

Prime numbers are not just of theoretical interest; they have numerous practical applications across a variety of fields:


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Cryptography: Prime numbers are fundamental to modern encryption algorithms, such as the RSA encryption system. In RSA, two large prime numbers are multiplied together to create a modulus, which is used for encrypting and decrypting sensitive information. The security of RSA and other cryptosystems relies on the difficulty of factoring large composite numbers into their prime factors.


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Engineering and Signal Processing: In engineering, prime numbers are used in signal processing and error detection algorithms. For example, the Fast Fourier Transform (FFT) relies on prime numbers to optimize signal analysis. Prime numbers also play a role in coding theory, which is used to detect and correct errors in data transmission.


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Computer Science: In hashing algorithms, prime numbers are used to distribute data evenly across a hash table, reducing collisions and improving performance. Prime numbers are also used in random number generation, which is essential for simulations, cryptography, and gaming.


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Quantum Computing: In quantum computing, prime numbers are essential for developing efficient algorithms to solve problems like integer factorization, which could revolutionize encryption and secure communications.


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Civil Engineering: Prime numbers are used in structural design and analysis, particularly in creating non-repetitive load distribution patterns that improve stability in buildings and bridges.


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Finance and Banking: Prime numbers are crucial in secure online transactions and the protection of financial data. They are used to generate keys in encryption algorithms that safeguard sensitive information in banking systems.



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Conclusion


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Prime numbers have fascinated mathematicians and scientists for millennia. From ancient Chinese assertions about their properties to modern discoveries in cryptography, signal processing, and quantum computing, prime numbers remain at the heart of many technological and scientific advancements. The study of prime numbers is not only a theoretical pursuit but also a practical one, influencing fields as diverse as engineering, finance, and computer science.


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As computing technology continues to advance, the search for larger prime numbers will likely continue, and new applications of prime numbers will emerge, further deepening our understanding of these remarkable mathematical objects.



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