Prime number
- See also: Math and the Bible
- See also: Best New Conservative Words
A prime number is a natural number greater than 1 which is divisible by only 1 and itself. Described another way, a prime number has only two factors, 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13 ....
As the building block for all numbers, prime numbers have a spiritual quality that has fascinated mathematicians and others. Prime numbers are a great way to learn and teach math, to improve mental acuity and focus, to fend off addiction, and to understand great works such as the Bible. The two greatest unsolved mysteries in mathematics relate to prime numbers: why every even number is the sum of two primes (Goldbach's conjecture), and apparently there are infinitely many twin primes. It is unknown whether there are infinitely many Mersenne primes.
The multiplication of the loaves -- the only miracle described in all four Gospels, and more than once in some Gospels, which suggests it happened more than once -- was always done with a prime number of loaves and fish (5 loaves and 2 fish, totaling 7, for the incidents described in Mark 6:31-44 , Matthew 14:19 , Luke 9:13 , and John 6:9 ). When another such incident is described later in the Gospel of Matthew, the multiplication occurs with another prime number-worth of loaves: 7. See Matthew 15:34 . Many biblical parables are tantalizing similar to prime numbers, such as The Parable of the Weeds. Acts 17:17, twice prime, is the midpoint of the New Testament.
In nature, multiple species of cicadas (locusts) reappear every 13 or 17 years -- both prime numbers -- and a full Moon on Christmas happens every 19 years, also a prime number. Humans have 23 pairs of chromosomes, a prime number.
Leonhard Euler wrote:Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.
There are infinitely many primes, while the only even prime number is 2. The Bible includes many references, direct and indirect, to prime numbers.
Contents
Sieve of Eratosthenes
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Here, n equals 100 |
To construct a table of all prime numbers less than n, you can use a method called the Sieve of Eratosthenes. Simply write down an ordered list of all counting numbers from 1 to n. Beginning with 2*2 = 4, cross out every second number:
- 2 3
4567891011 ...
then beginning with 3*3 = 9 cross out every third number:
- 2 3
45678 9 1011 ...
Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until .
The number of primes smaller than a given number is roughly , where is the natural logarithm (base e) of .^{[1]} This is a formulation of a more general statement known as the Prime Number Theorem.
How many prime numbers
For a more detailed treatment, see Prime Number Theorem.
It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set—there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes (by the Fundamental Theorem of Arithmetic), N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.
Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):
The largest known prime is 2^{57,885,161}-1, a 17,425,169 digit number beginning with 581,887,266,232... and ending with ...951; it was discovered in 2013 by Curtis Cooper with the Great Internet Mersenne Prime Search (GIMPS),^{[2]} a distributed computing project launched by George Woltman in early 1996.^{[3]}
A Mersenne prime is a prime number M that is of the form M = 2^{n} − 1, where n is some prime number.^{[4]} As of 2015, only 48 Mersenne Primes have been found.^{[5]} The largest known prime, 2^{57885161}-1, is a Mersenne Prime. The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician Marin Mersenne.
Unique factorization
For a more detailed treatment, see Fundamental Theorem of Arithmetic.
According to the Fundamental Theorem of Arithmetic, proven by Carl Friedrich Gauss, every positive integer has a unique factorization into prime numbers.
This means that every integer larger than 1 can be expressed as a product of one or more primes in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.
Finding the prime factors for large numbers (having hundreds of digits) can take considerable time (millions of years) even with the most advanced computers.
Alternative definition
Beyond the integers, mathematicians are interested in other structures with addition and multiplication; they are called rings. When working in rings, mathematicians use a different definition for a "prime" element:A prime element is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.
Or in symbolic notation:
For the integers, this definition is equivalent to the one stated above. However, in general the two definitions are not equivalent. The earlier definition is still important in rings and is given the name irreducibility, since it literally states that the element cannot be broken into smaller pieces.
The preference for this alternate definition of "prime" is that this definition gives you unique factorization theorems analogous to the Fundamental Theorem of Arithmetic. For more information, consult an abstract algebra book^{[6]}
Primality testing
To determine whether a number is prime, you can use the trial division method, provided the number you are testing is not too large. Let N be the number being tested and s be its square root. Divide N by each number in the prime table, beginning with 2. If there is no remainder, stop--N is composite. Otherwise, test again with the next largest prime in the table. If the next largest prime is greater than s, stop--N is prime.
For larger values of N, a method based on Fermat's theorem can be used, though it usually requires a computer. Start with the number 1, and double it N-1 times. Divide this number by N, keeping only the remainder. If the remainder is greater than 1, N is composite. Unfortunately, the converse is not true—if the remainder is 1, N is probably prime, but not necessarily. Algorithms that improve on this method exist, and primality testing is today an active subject of mathematical research.
Open problems about prime numbers
Twin primes
For a more detailed treatment, see twin primes.
Twin primes or prime twins are pairs of prime numbers that differ from each other by only 2. Since all primes other than 2 are odd numbers, this is the closest two primes can be to each other, with the exception of 2 and 3. The first few twin primes are 3 and 5, 5 and 7, 11 and 13, 17 and 19, and 29 and 31.
Twin primes become increasingly scarce among larger numbers. For example, there are twenty-four sets of twin primes between 0 and 500, but only eleven sets between 500 and 1000. Computer programs have calculated some extremely large sets of twin primes. While there is undoubtedly an unlimited number of primes, opinions differ among mathematicians on whether there is also an infinite number of twin primes or whether there is an upper limit. This debate is captured by the twin primes conjecture
Mathematicians have long been fascinated by twin primes and the relationship between them. There seems to be no distinct pattern in their occurrence. For example, there are no twin primes between 700 and 800, or between 900 and 1000, but there are five sets between 800 and 900 (809 and 811; 821 and 823; 827 and 829; 857 and 859; 881 and 883).
Goldbach's conjecture
Goldbach's conjecture asserts that every even integer greater than two is the sum of two prime numbers. For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.
This conjecture is one of the oldest unsettled statements in mathematics. For centuries mathematicians have attempted to prove this statement, without success. The conjecture is known to hold for numbers up to at least 3 10^{17}.
Riemann hypothesis
For a more detailed treatment, see Riemann hypothesis.
The Riemann hypothesis is related to patterns in the distribution of prime numbers. The Clay Mathematics Institute has offered a one million dollar prize for proof of the Riemann hypothesis.^{[7]}
Cryptography
The security of public-key cryptography schemes such as RSA depends on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such cryptographic schemes could become vulnerable to attack.
Composite numbers
The opposites of a prime number are a composite number and a highly composite number. A composite number is the general opposite to a prime number. And a highly composite number is the special, or extreme, opposite to a prime number.
Quotations
“ | God may not play dice with the universe, but something strange is going on with the prime numbers. - Paul Erdos^{[8]} | ” |
“ | Prime numbers are what is left when you have taken all the patterns away. - The Curious Incident of the Dog in the Night-Time, by Mark Haddon^{[9]} | ” |
“ | The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. - Carl Gauss^{[10]} | ” |
See also
- Composite number
- Essay:Best New Conservative Words - akin to prime numbers
- Fibonacci sequence
- Math and the Bible
- Math and God
- Perfect number
External links
- 20 Best Prime Numbers Books of All Time
- prime number game
- Great Internet Mersenne Prime Search project website
- Prime Number resource Archive
- Sieve of Eratosthenes
References
- ↑ http://home.att.net/~numericana/answer/primes.htm
- ↑ http://www.mersenne.org/
- ↑ Multiple references:
- http://primes.utm.edu/largest.html
- 2^{57,885,161} = 10 ^ (57,885,161 log_{10} 2). The number two raised to successively larger powers causes the last three digits to fairly quickly begin repeating themselves in cycles of one hundred, so it is known that the last digits of this Mersenne prime end with ...951. Written out, the number would be Five hundred eighty-one quinmilliamilliaoctingenoctomilliatrecenoctooctogintillion eight hundred eighty-seven quinmilliamilliaoctingenoctomilliatrecenseptenoctogintillion etc.
- ↑ Mersenne Prime from Wolfram
- ↑ http://primes.utm.edu/largest.html#largest
- ↑ Abstract Algebra: An Introduction, Thomas Hungerford. Brooks Cole; 2 edition (1996)
- ↑ http://www.claymath.org/millennium/Riemann_Hypothesis/
- ↑ https://www.brainyquote.com/quotes/paul_erdos_342886
- ↑ https://www.bookbrowse.com/excerpts/index.cfm/book_number/1252/page_number/5/the-curious-incident-of-the-dog-in-the-nighttime
- ↑ https://www.azquotes.com/quote/916300?ref=prime-numbers