Download Math 105 Worksheet: The Infinitude of Primes 1. Let Qn = p1p2···pn

Math 105
Worksheet: The Infinitude of Primes
1. Let Qn = p1p2···pn, the product of the first n primes. For example,
Q1 = 2
Q2 = (2)(3) = 6
Q3 = (2)(3)(5) = 30
Q4 = __________
Q5 = __________
2. Write down the value of Q4 + 1 here: ________________
a. What is the remainder when Q4 + 1 is divided by 2?
b. What is the remainder when Q4 + 1 is divided by 3?
c. What is the remainder when Q4 + 1 is divided by 5?
d. What is the remainder when Q4 + 1 is divided by 7?
p1 = 2
p2 = 3
p3 = 5
p4 = 7
p5 = 11
p6 = 13
p7 = 17
p8 = 19
p9 = 23
p10 = 29
p11 = 31
p12 = 37
p13 = 41
p14 = 43
p15 = 47
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.
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3. Now consider the number Q14 + 1. (You don’t actually have to compute this number. Just
think about it.)
a. True or False: 19 is a divisor of Q14 + 1. Explain.
b. What is the remainder if you divide Q14 + 1 by 19?
4. Explain why none of the primes 2, 3, 5, 7, 11, …, 43 can be divisors of Q14 + 1.
5. Suppose you find the prime factorization of Q14 + 1. What is the smallest possible prime
factor of Q14 + 1 ?
6. Your annoying know-it-all little sister claims that there are only finitely many prime
numbers, namely, p1, p2, p3, …. , pn, for some number n. Consequently, she says, any
number, no matter how large, is a product of these primes.
Use the number Qn + 1 to convince your sister that she is wrong. Write a few sentences
below expressing your reasoning.