Opening Exercise: When is 𝟐𝐧 − 𝟏 prime and when is it composite? Complete the table to investigate which numbers of the form 2𝑛 −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 𝑚 is a positive odd composite number, then 2𝑚 − 1 is a composite number.
Start with an identity: 𝑥𝑛 − 1 = (𝑥 − 1)(𝑥𝑛-1 + 𝑥𝑛-2 + ⋯ 𝑥1 + 1)
In this case, 𝑥 = 2, so the identity above becomes: 2𝑚 − 1 = (2 − 1)(2𝑚-1 +2𝑚-2 + ⋯ + 21 + 1) = (2𝑚-1 + 2𝑚-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 2𝑚 − 1 = (2𝑎)𝑏 − 1 = (2𝑎 − 1) ((2𝑎)𝑏-1 +(2𝑎)𝑏-2 + ⋯ +(2𝑎) ⏟ 1 + 1 )
Since 𝑎 ≥ 3, we have 2𝑎 ≥ 8; thus, 2𝑎 − 1 ≥ 7. Since the other factor is also larger than 1, 2𝑚 − 1 is composite, and we have proven our conjecture.
For Exercises 1–3, find a factor of each expression using the method discussed in Example 1.
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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