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Common Core For Algebra

Student Outcomes

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

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 2^{π} β1 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:
2^{π} β 1 = (2 β 1)(2^{π-1} +2^{π-2} + β― + 21 + 1)
= (2^{π-1} + 2^{π-2} + β― +21 + 1),
and it is not clear whether or not 2π β1 is composite.

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

Then we have
2^{π} β 1 = (2^{π})^{π} β 1
= (2^{π} β 1) ((2^{π})^{π-1} +(2^{π})^{π-2} + β― +(2^{π}) β 1 + 1 )

Since π β₯ 3, we have 2^{π} β₯ 8; thus, 2^{π} β 1 β₯ 7. Since the other factor is also larger than 1, 2^{π} β 1 is
composite, and we have proven our conjecture.

Exercises 1β3

For Exercises 1β3, find a factor of each expression using the method discussed in Example 1.

- 2
^{15}β 1 - 2
^{99}β 1 - 2
^{537}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^{(πβ13)/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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