Class #6

Erdös: The Man Who Loved Only Numbers

Assignments for Class #6

• Read Chapter 4, pages 179-201

• Watch this video, about Fibonacci numbers, and their occurrence in nature:

Questions to Think About

• In your garden or your plant pots, see you if can find any examples of flowers, seed heads, or other plant parts that grow in Fibonacci numbers of spirals, or with Fibonacci numbers of objects in each spiral. 

Let Vihart help you to see them:

• Think back to relatively prime numbers. The topic is not in the index of the text, but see pages 38 and 132 (I added these entries to the index of my copy.) 

(Remember that 1 is not a prime number. Some people call 0 and 1 pre-primes, meaning that they come before the "real" primes. Some people include 2 in the pre-primes; the term pre-prime is informal, not a common technical term.)

Two integers that have no prime factors in common are called relative primes. The first pair of relative primes are 2 and 3.  (correct?)

    - What is the next pair of relative primes after 2 and 3?
    - Do all pairs of primes make relative pairs?
    - Are there two even numbers that make a relative pair?
    - What is the first relative pair in which one number is prime and one is composite?
    - What is the first pair of composite numbers that are relative primes?

• What is the greatest common divisor of the integers 24 and 54? Of what use is the greatest common divisor in everyday arithmetic? Learn more about greatest common divisor at Wikipedia.

• You are in a foreign country with strange currency. You are holding a pile of coins of 2 varieties: a 12 cent coin and a 54 cent coin.   You have many coins of each variety in the pile.  Can you use your coins to pay a bill of $4.55 exactly? If so, how? If not, why not? Challenging questions: What about $4.58?  or $4.62?  What bill amounts can work and which cannot?

• Which of these graphs below are traversable, meaning: can you trace a path all the way around the graph using all lines exactly once, starting at a node and ending up either there or at some other node?


Other Resources

• More about Fibonacci numbers at Wikipedia

• Learn more about Fibonacci spirals from Vihart's next two videos on the subject:


• Remember that a rational number is one that can be expressed as a ratio of integers (think rational). π is irrational, while 22/7 is very close to π, but it's rational. Can one number be more irrational than another? Here's a video about the golden ratio, and what makes it so irrational:


• Basics and more about 𝞿, the golden ratio, at Wikpedia

• Just in time to connect with our course material in Chapter 4 on Diophantine equations, this article appeared last week (14 October 2021) at Scientific American: "Gnarly, Centuries-Old Mathematical Quandaries Get New Solutions." Included in the first paragraph is a link to another article that might interest you, about unsolved puzzles in mathematics.

• Here is an eleven-minute long interview with Jim Holt, the author of When Einstein Walked with Gödel.  The section about Gödel starts about 4 minutes in, but the entire interview includes a discussion of Alan Turing, Georg Cantor, and Evariste Galois as well.

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