The Power of Algebraβ€”Finding Primes

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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 2


Opening Exercise: When is 𝟐𝐧 βˆ’ 𝟏 prime and when is it composite? Complete the table to investigate which numbers of the form 2𝑛 βˆ’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‘š is a positive odd composite number, then 2π‘š βˆ’ 1 is a composite number.

Start with an identity: π‘₯𝑛 βˆ’ 1 = (π‘₯ βˆ’ 1)(π‘₯𝑛-1 + π‘₯𝑛-2 + β‹― π‘₯1 + 1)

In this case, π‘₯ = 2, so the identity above becomes: 2π‘š βˆ’ 1 = (2 βˆ’ 1)(2π‘š-1 +2π‘š-2 + β‹― + 21 + 1) = (2π‘š-1 + 2π‘š-2 + β‹― +21 + 1), and it is not clear whether or not 2π‘š βˆ’1 is composite.

Rewrite the expression: Let π‘š = π‘Žπ‘ be a positive odd composite number. Then π‘Ž and 𝑏 must also be odd, or else the product π‘Žπ‘ would be even. The smallest such number π‘š is 9, so we have π‘Ž β‰₯ 3 and 𝑏 β‰₯ 3.

Then we have 2π‘š βˆ’ 1 = (2π‘Ž)𝑏 βˆ’ 1 = (2π‘Ž βˆ’ 1) ((2π‘Ž)𝑏-1 +(2π‘Ž)𝑏-2 + β‹― +(2π‘Ž) ⏟ 1 + 1 )

Since π‘Ž β‰₯ 3, we have 2π‘Ž β‰₯ 8; thus, 2π‘Ž βˆ’ 1 β‰₯ 7. Since the other factor is also larger than 1, 2π‘š βˆ’ 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 βˆ’ 1
  2. 299 βˆ’ 1
  3. 2537 βˆ’ 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 Γ— 10(π‘›βˆ’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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