# The Power of AlgebraβFinding Primes

### The Power of Algebraβ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 II, Module 1, Lesson 8

Classwork

Opening Exercise: When is ππ§ β π prime and when is it composite? Complete the table to investigate which numbers of the form 2π β1 are prime and which are composite

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

Example 1: Proving a Conjecture

Conjecture: If π is a positive odd composite number, then 2π β 1 is a composite number.

Start with an identity: π₯π β 1 = (π₯ β 1)(π₯π-1 + π₯π-2 + β― π₯1 + 1)

In this case, π₯ = 2, so the identity above becomes: 2π β 1 = (2 β 1)(2π-1 +2π-2 + β― + 21 + 1) = (2π-1 + 2π-2 + β― +21 + 1), and it is not clear whether or not 2π β1 is composite.

Rewrite the expression: Let π = ππ be a positive odd composite number. Then π and π must also be odd, or else the product ππ would be even. The smallest such number π is 9, so we have π β₯ 3 and π β₯ 3.

Then we have 2π β 1 = (2π)π β 1 = (2π β 1) ((2π)π-1 +(2π)π-2 + β― +(2π) β 1 + 1 )

Since π β₯ 3, we have 2π β₯ 8; thus, 2π β 1 β₯ 7. Since the other factor is also larger than 1, 2π β 1 is composite, and we have proven our conjecture.

Exercises 1β3

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

1. 215 β 1
2. 299 β 1
3. 2537 β 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 π-digit number of the form π2, where π is a large prime, can roughly be approximated by π(π) = 3.4 Γ 10(πβ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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