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TExES Physics/Mathematics 7-12 (243): Practice & Study Guide62 chapters | 688 lessons | 60 flashcard sets

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Although computers facilitate searches for prime numbers and perfect numbers, fundamental number theory is still very relevant. In this lesson we explore how a particular type of prime number, the Mersenne prime, relates to perfect numbers.

Searching for perfect numbers and prime numbers began thousands of years ago and continues to this day. Perfect numbers and prime numbers are uniquely defined by their divisors. Some of the prime numbers discovered are Mersenne primes. These primes are linked to powers of 2. In this lesson we explore Mersenne primes and their fascinating relationship with perfect numbers.

Just for fun, add the divisors of 6 not including the 6. Adding 1 plus 2 plus 3 gives 6. Six is the starting number and 6 is the sum. Six is a **perfect number** where the sum of the divisors (not including the number itself) equals the number.

What about the number 8?

First, find the divisors of 8. The divisors of 8 are 1, 2, 4 and 8. Is 8 a perfect number? Adding 1 + 2 + 4 gives 7 which does not equal 8. Thus, 8 is not a perfect number.

How about 3? The divisors of 3 are 1 and 3. Not a perfect number either but 3 is a **prime number**, in that it has only two divisors: the number itself and 1.

Could a prime number ever be a perfect number? What is the divisor sum if the number is a prime number? Right, always a 1. Thus, none of the prime numbers are perfect numbers.

Ready for another idea? There's a special kind of prime number called the Mersenne prime.

**Mersenne primes** are prime numbers that are one less than a power of 2; compactly written as 2*n* - 1. A systematic way to find Mersenne primes:

- calculate 2
*n*for*n*= 1, 2, 3, .... - subtract 1
- check for prime numbers

Instead of doing a thorough search right away, let's do *n* = 2.

Step 1, calculate 2*n* for *n* = 2. Thus, 22 = 2(2) = 4.

Step 2, subtract 1 from 4 which is 3.

Step 3, check if 3 is prime. The divisors of 3 are only 3 and 1. Thus, 3 is prime.

Conclusion: 3 is a Mersenne prime.

Do all prime numbers have the 2*n* - 1 recipe? How about the prime number 5? Well, 5 is prime because the divisors are 5 and 1. But there no *n* for the 2*n* - 1 would result in 5. Although 5 is prime number, 5 is not a Mersenne prime.

Ready for another idea? Mersenne primes can be used to find perfect numbers.

Having a Mersenne prime, the *n* is known. With this *n* we find a perfect number using the formula 2*n*-1 (2*n* - 1). This formula contains the 2*n* - 1 Mersenne primes with a 2*n*-1 multiplier in front.

Remember the perfect number, 6, and the Mersenne prime, 3? The *n* was 2 in 2*n* - 1 to get the 3. Now, use this *n* = 2 in 2*n*-1 (2*n* - 1) to get a perfect number.

22-1 (22 - 1) = 21 (4 - 1) = 2(3) = 6. There's our perfect number, 6.

Let's find the first 3 Mersenne primes using a systematic search:

- Write some increasing values for
*n*starting with*n*= 1:

- Calculate 2
*n*for each of these values for*n*:

For example, for *n* = 5, 25 is 2(2)(2)(2)(2) = 2(4)(4) = 2(16) = 32.

- Subtract 1 from each result:

- From this list, pick out the prime numbers:

3 7 31 … are the prime numbers. The number 15 has more than two divisors, and so is not prime. Note: 1 is not prime because it has only one divisor.

Thus, the first three Mersenne primes are 3, 7 and 31 corresponding to *n* = 2, 3 and 5. Another observation is the *n* for Mersenne primes are prime numbers themselves. However, not all prime numbers will yield a Mersenne prime. For example, *n* = 11 is prime but 211 - 1 is 2047 which is not a prime number. The divisors of 2047 are 1, 23, 89 and 2047.

Now, for the perfect numbers.

- For
*n*= 2, 3 and 5, calculate 2*n*-1 (2*n*- 1)

We recognize the perfect number, 6. How about 28? Let's do the details for the perfect number computation leading to 28. For *n* = 3, 2*n*-1 = 23 - 1 = 2(2)(2) - 1 = 8 - 1 = 7. This is the Mersenne prime. For the factor in front, 2*n*-1 = 23-1 = 22 = 4. So, the perfect number is 2*n*-1 (2*n* - 1) = 4(7) = 28. Adding the divisors: 1 + 2 + 4 + 7 + 14 = 7 + 21 = 28. It works!

If you feel like checking 496, the divisors are 1, 2, 4, 8, 16, 31, 62, 124 and 248. Does this list add to 496? Answer: yes!

A number with only two divisors (itself and 1) is a **prime number**. If a prime number can be written as 2*n* - 1 for some *n*, the prime number is a **Mersenne prime**. If the sum of divisors of a number (excluding the number itself) equals the number, the number is a **perfect number**. Perfect numbers are related to Mersenne primes. To find a perfect number, calculate 2*n*-1 (2*n* - 1) where *n* is the number used to obtain a Mersenne prime.

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TExES Physics/Mathematics 7-12 (243): Practice & Study Guide62 chapters | 688 lessons | 60 flashcard sets

- What Are Prime Numbers? - Definition & Examples 5:03
- What Are Composite Numbers? - Definition & Examples 4:38
- What Are Odd & Even Numbers? - Definition & Examples 6:38
- How to Find the Prime Factorization of a Number 5:36
- Finding the Prime Factorization with Exponents 4:44
- Using Prime Factorizations to Find the Least Common Multiples 7:28
- Word Problems: Greatest Common Factor & Least Common Multiple 4:34
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- Perfect Numbers & Mersenne Primes
- Fundamental Theorem of Algebra: Explanation and Example 7:39
- The Fundamental Theorem of Calculus 7:52
- Euclidean Algorithm & Diophantine Equation: Examples & Solutions 7:01
- Go to TExES Physics/Math 7-12: Number Theory

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