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.

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