Primes, Factors, and Perfect Numbers Presented by Dr. Lisa Carnell High Point University lcarnell@highpoint.edu
Presented at the North Carolina Council of Teachers of Mathematics State Conference October 11-12, 2007 Greensboro, NC
Prime Numbers Definition: A prime number is a positive integer greater than one that is divisible by no positive integers except one and itself. The first twelve prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37. History • Primes and their properties were first studied extensively by the ancient Greeks. • Those in the Pythagoras’s school were interested in the mystical and numerological properties of numbers. • Euclid’s Elements (ca. 300 BC) contained important results about primes including proofs there are infinitely many primes and that every integer can be written as a unique product of primes (Fundamental Theorem of Arithmetic). • Eratosthenes (ca. 200 BC) devised a method for calculating primes called the Sieve of Eratosthenes. • The next major development on primes was almost 1800 years later when Fermat, at the beginning of the 17th century proved that every prime of the form 4n + 1 can be written in a unique way as the sum of two squares. Example: 5 = 22 + 12, 13 = 32 + 22 • Fermat also devised a new way for factoring large numbers which forms the basis for checking whether a number is prime and is still used in computer algorithms. • Before computers, an early search for large primes was done by Pietro Cataldi who showed by 1588 that 131,071 and 524,287 were prime. By 1772 Euler showed that 2,147,483,647 was prime. • With the advent of computers, people spend time trying to find the largest known prime. As of September 2006, the largest known prime is the 9,808,358 digit prime 232582657 – 1. • Since it has been proven that there are infinitely many primes, the race to find the next largest one will continue. We know there are infinitely many primes because if you multiply all the known primes together and add 1, then you would get a number that must be divisible by at least one new prime number. • By using regression analysis, it is predicted that a one-billion digit prime will be found by 2024.
Activity 1: Have students guess how many digits the largest known prime has. Estimate how long it would take to write out that prime by hand. Estimate the length of a strip of paper with that number written on it. Activity 2: Use the Sieve of Eratosthenes to find all the primes less than 100. For a companion activity, see Eratosthenes’ Sieve www.keypress.com/documents/ALookInside/MathExplorer/MathExplorer_p p_46_54.pdfActivity 3: Identify prime numbers, composite numbers, and factors of composite numbers by forming arrays with manipulatives. For a companion activity, see Prime and Composite Numbers With Use of Manipulatives http://www.teachers.net/lessons/posts/2955.html Prime, Composite, and Square Numbers http://www2/ups/edu/community/tofu/lev2/mathconcepts/factprimefunct/fac tors.htm Activity 4: The numbers 2 and 3 are consecutive prime numbers. Are there any other consecutive prime numbers? Activity 5: Twin primes are primes that are separated by two. Example: 5 and 7. The Twin Primes Conjecture says there are infinitely many twin primes. Find all the twin primes less than 100. Activity 6: Goldbach’s Conjecture says that every even number greater than 2 can be written as the sum of two (not necessarily distinct) primes. Example: 12 = 5 + 7. Verify this conjecture for even integers 4 to 24. Another of Goldbach’s conjectures was that every integer greater than 5 is the sum of three prime numbers. Verify this conjecture for integers 6 through 24.
Perfect Numbers Definition: A perfect number is a positive integer that is equal to the sum of its proper divisors. That is, the number itself is not included in the sum. Example: 6 = 1 + 2 + 3 The first four perfect numbers are 6, 28, 496, and 8128, and these have been known since ancient times. In December of 2001 a perfect number was found that had over 4 million digits. Square Numbers Definition: Square numbers are numbers that can be expressed as the product of two equal factors. Example: 25 = 5x5 The first five square numbers are 1, 4, 9, 16, and 25.
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Activity 7: Form the first 10 square numbers. Pick any two consecutive square numbers and subtract them (larger – smaller). What do you notice about the difference? What do you notice about the sequence of the consecutive differences? Triangular Numbers Definition: Triangular numbers are numbers of the form n(n + 1)/2 for n = 1, 2, 3, ‌ * * * * * ** *** 1 3 6 Activity 8: Form the first 10 triangular numbers. Pick any two consecutive triangular numbers and add them together. What do you notice about the sum?
Amicable Numbers Definition: Amicable numbers are a pair of numbers with the property that the sum of all the proper divisors of the first number equals the second number while the sum of the proper divisors of the second number equals the first number. Example: Divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 The sum of the divisors of 220 is 284. Divisors of 284: 1, 2, 4, 71, 142 The sum of the divisors of 284 is 220. So 220 and 284 are amicable numbers. The first four amicable pairs are 220 and 284, 1184 and 1210, 2620 and 2924, 5020 and 5564. Today there are over 5000 known pairs of amicable numbers. Deficient Numbers Definition: A number is deficient if the sum of the proper divisors of the number is smaller than the number itself. Abundant Numbers Definition: A number is abundant if the sum of the proper divisors of the number is larger than the number itself. Activity 9: Identify the numbers from 2 to 30 as deficient numbers or abundant numbers. Are prime numbers deficient or abundant? Explain. Fibonacci Numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ‌ The sequence is formed by adding the two previous numbers to get the next one. The Fibonacci sequence is found throughout nature in sea shell spirals, petals on flowers, and pine cones. Activity 10: If a Fibonacci number is divided by the previous number in the sequence, what happens to the values of the quotient?
Activity 11: Using a seed catalog or website, identify flowers that have a number of petals that are Fibonacci numbers. For a companion activity see The Fibonacci Numbers and Golden Section in Nature http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#petals Activity 12: Design your own class of numbers. Make a poster explaining the properties of your numbers. Divisibility Rules 2: If the last digit is even, the number is divisible by 2. 3: If the sum of the digits is divisible by 3, then so is the number. 4: If the number formed by the last two digits is divisible by 4, then so is the number. 5: If the last digit is a 0 or 5, the number is divisible by 5. 6: If the number is divisible by 2 and 3, the number is divisible by 6. 8: If the last three digits form a number divisible by 8, then the whole number is divisible by 8. 9: If the sum of the digits is divisible by 9, then the number is divisible by 9. 10: If the number ends in 0, it is divisible by 10. Activity 13: If a number is divisible by 3 and 4, is it divisible by 12? If a number is divisible by 2 and 6, is it divisible by 12? Explain. Activity 14: Create your own divisibility rule for 15. For a companion activity, see Divisibility War http://edweb.sdsu.edu/courses/edtec670/Cardboard/card/d/Divisibility.html
ANSWERS: Activity 1: As of September 2006, the largest known prime is the 9,808,358 digit prime 232582657 – 1. Estimates will vary. Activity 2: See hundreds chart. Activity 3: Consider 12 as an example. Possible arrays are 1x12, 2x6, 3x4 indicating the factors of 12 are 1, 2, 3, 4, and 12. Activity 4: No, because consecutive numbers have one even number and one odd number, and 2 is the only even prime. Activity 5: 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73 Activity 6: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, and so on. 6 = 2 + 2 + 2, 7 = 2 + 2 + 3, 8 = 2 + 3 + 3, and so on. Activity 7: The difference is always odd. The consecutive differences are consecutive odd numbers. Activity 8: The sum is a square number. Activity 9: Prime numbers are deficient since the only proper divisor they have is 1. Activity 10: The values start to settle down to the Golden Ratio, about 1.618034. Activity 11: Answers will vary. Activity 12: Answers will vary. Activity 13: Numbers divisible by 3 and 4 are divisible by 12. Numbers divisible by 2 and 6 are not necessarily divisible by 12 (example: 18). When a positive integer (i.e. 12) can be written as the product of two numbers (i.e. 3 and 4) which have no common factor bigger than 1 (making them relatively prime), testing divisibility by that number can be reduced to testing divisibility by the factors.
Activity 14: If a number is divisible by 3 and 5, it is divisible by 15. Resources Eratosthenes’ Sieve www.keypress.com/documents/ALookInside/MathExplorer/MathExplorer_pp_46_54.pdf Divisibility War http://edweb.sdsu.edu/courses/edtec670/Cardboard/card/d/Divisibility.html The Fibonacci Numbers and Golden Section in Nature http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#petals Largest Known Prime by Year: A Brief History http://primes.utm.edu/notes/by_year.html Mudd Math Fun Facts http://www.math.hmc.edu/funfacts Numbers from one to thirty-one‌a page for each day of the month http://richardphillips.org.uk/number/Num11.htm. Perfect Numbers http://www-history.mcs.st-and.ac.uk/history/HistTopics/Perfect_numbers.html Perfect Number et al. http://trottermath.net/numthry/perfnos.html Prime and Composite Numbers with use of manipulatives http://www.teachers.net/lessons/posts/2955.html Prime, Composite, and Square Numbers http://www2/ups/edu/community/tofu/lev2/mathconcepts/factprimefunct/factors.htm Prime Numbers http://mathforum.org/isaac/problems/prime1.html Prime Numbers http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html Topics About Special Numbers http://www.math.wichita.edu/history/topics/snumbers.html