220 CHAPTER 5 Number Theory and the Real Number System More About Prime Numbers More than 2000 years ago, the Greek mathematician Euclid proved that there is no largest prime number. Mathematicians, however, continue to strive to discover larger and larger prime numbers. Marin Mersenne (1588–1648), a seventeenth-century monk, discovered that numbers of the form 2 1 n − are often prime numbers when n is a prime number. For example, 2 1 4 1 3 2 1 8 1 7 2 3 − = − = − = − = 2 1 32 1 31 2 11281127 5 7 − = − = − = − = Numbers of the form 2 1 n − that are prime are referred to as Mersenne primes . The first 10 Mersenne primes occur when n 2, = 3, 5, 7, 13, 17, 19, 31, 61, 89. The first time the expression 2 1 n − does not generate a prime number, for prime number n, is when n is 11. The number 2 1 11 − is a composite number (see Exercise 90). Scientists frequently use Mersenne primes in their search for larger and larger primes. The largest prime number found to date was discovered on December 7, 2018, by Patrick Laroche of Ocala, Florida. Laroche worked in conjunction with the Great Internet Mersenne Prime Search (GIMPS; see Mathematics Today, at left) to conduct the search. The number is Mersenne prime 2 1. 82,589,933 − This record prime number is the fifty-first known Mersenne prime. When written out, it is 24,862,048 digits long. When written out using a standard 12-point font, the number is almost 33 miles long! Another mathematician who studied prime numbers was Pierre de Fermat (1601– 1665). A lawyer by profession, Fermat became interested in mathematics as a hobby. He became one of the finest mathematicians of the seventeenth century. Fermat conjectured that each number of the form 2 1, 2n + now referred to as a Fermat number , was prime for each natural number n. Recall that a conjecture is a hypothesis that has not been proved or disproved. In 1732, Leonhard Euler proved that for n 5, 2 1 2 1 2 32 5 = + = + was a composite number, thus disproving Fermat’s conjecture. Since Euler’s time, mathematicians have been able to evaluate only the sixth, seventh, eighth, ninth, tenth, and eleventh Fermat numbers to determine whether they are prime or composite. Each of these numbers has been shown to be composite. The eleventh Fermat number was factored by Richard Brent and François Morain in 1988. The sheer magnitude of the numbers involved makes it difficult to test these numbers, even with supercomputers. In 1742, German mathematician Christian Goldbach conjectured in a letter to Euler that every even number greater than or equal to 4 can be represented as the sum of two (not necessarily distinct) prime numbers (for example, 4 2 2, = + 6 33,8 35,10 55,12 57 = + = + = + = + ). This conjecture became known as Goldbach’s conjecture , and it remains unproven to this day. The twin prime conjecture is another famous long-standing conjecture. Twin primes are primes of the form p and p 2 + (for example, 3 and 5, 5 and 7, 11 and 13). This conjecture states that there are an infinite number of pairs of twin primes. At the time of this writing, the largest known twin primes are of the form 2,996,863,034,895 2 1, 1,290,000 ⋅ ± which were discovered by a collaborative effort of two research groups, Twin Prime Search and PrimeGrid , on September 15, 2016, and contains 388,342 digits. MATHEMATICS TODAY GIMPS The Great Internet Mersenne Prime Search (GIMPS) was started by George Woltman in 1996 to coordinate the computing efforts of thousands of people interested in determining new Mersenne prime numbers. Anyone with a computer and Internet access can join the project. GIMPS uses idle time on the participants’ computers to search for prime numbers. As of this writing, the largest prime number discovered so far was discovered by Patrick Laroche of Ocala, Florida, in conjunction with GIMPS on December 7, 2018. The number, 2 1, 82,589,933 − almost 25 million digits long, was the fiftyfirst known Mersenne prime number and the seventeenth found by GIMPS participants. Why This Is Important Determining larger and larger prime numbers is a major part of the history of mathematics that continues to this day. Prime numbers are used in encryption to help keep financial transactions secure. Exercises Warm Up Exercises In Exercises 1–10, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The study of numbers and their properties is known as number __________. Theory 2. If a is a factor of b, then b a ÷ is a(n) __________. Integer 3. If a is divisible by b, then a b ÷ has a remainder of __________. Zero 4. A natural number greater than 1 that has exactly two factors, itself and 1, is known as a(n) __________ number. Prime SECTION 5.1 Learning Catalytics Keyword: Angel-SOM-5.1 (See Preface for additional details.)
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