The Power of Algebra—Finding Primes

The Power of Algebra—Finding Primes

Student Outcomes

• Students apply polynomial identities to the detection of prime numbers.

New York State Common Core Math Algebra II, Module 1, Lesson 8

Worksheets for Algebra 2

Classwork

Opening Exercise: When is 𝟐^𝐧 − 𝟏 prime and when is it composite? Complete the table to investigate which numbers of the form 2^𝑛 −1 are prime and which are composite

What patterns do you notice in this table about which expressions are prime and which are composite?

• Show Video for the Lesson

Example 1: Proving a Conjecture

Conjecture: If 𝑚 is a positive odd composite number, then 2𝑚 − 1 is a composite number.

Start with an identity: 𝑥^𝑛 − 1 = (𝑥 − 1)(𝑥^𝑛-1 + 𝑥^𝑛-2 + ⋯ 𝑥¹ + 1)

In this case, 𝑥 = 2, so the identity above becomes: 2^𝑚 − 1 = (2 − 1)(2^𝑚-1 +2^𝑚-2 + ⋯ + 21 + 1) = (2^𝑚-1 + 2^𝑚-2 + ⋯ +21 + 1), and it is not clear whether or not 2𝑚 −1 is composite.

Rewrite the expression: Let 𝑚 = 𝑎𝑏 be a positive odd composite number. Then 𝑎 and 𝑏 must also be odd, or else the product 𝑎𝑏 would be even. The smallest such number 𝑚 is 9, so we have 𝑎 ≥ 3 and 𝑏 ≥ 3.

Then we have 2^𝑚 − 1 = (2^𝑎)^𝑏 − 1 = (2^𝑎 − 1) ((2^𝑎)^𝑏-1 +(2^𝑎)^𝑏-2 + ⋯ +(2^𝑎) ⏟ 1 + 1 )

Since 𝑎 ≥ 3, we have 2^𝑎 ≥ 8; thus, 2^𝑎 − 1 ≥ 7. Since the other factor is also larger than 1, 2^𝑚 − 1 is composite, and we have proven our conjecture.

Exercises 1–3

For Exercises 1–3, find a factor of each expression using the method discussed in Example 1.

1. 2¹⁵ − 1
2. 2⁹⁹ − 1
3. 2⁵³⁷ − 1 (Hint: 537 is the product of two prime numbers that are both less than 50.)

Exercise 4: How quickly can a computer factor a very large number? 4. How long would it take a computer to factor some squares of very large prime numbers? The time in seconds required to factor an 𝑛-digit number of the form 𝑝², where 𝑝 is a large prime, can roughly be approximated by 𝑓(𝑛) = 3.4 × 10^{(𝑛−13)/2}. Some values of this function are listed in the table below.

Use the function given above to determine how long it would take this computer to factor a number that contains 32 digits.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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