Farey Sequence

Goals

• Explore patterns in irreducible fractions

• Calculate Euler's Phi function

What to Do

• 1. Get student to write out all fractions between 0 and 1 where the denominator is 5 or less

  2. Remove all reducible fractions from the list. (Explain the numerator and denominator are called co-prime if the fraction is irreducible.)

  3. Then get them to put the fractions in value order, lowest to highest. Explain how to do this if necessary

  4. Next write out the fraction is a row. Consider group of 3 adjacent fractions. Add the numerator and denominator together. Are there any common patterns.

  5. Find Farey sequence and use this to add more fractions with higher denominators, upto at least 10.

  6. Part 2: For each of the denominators, count the number of irreducible fractions between 0 and 1. Write down the number of fraction for each denominator. Can you find any patterns in the numbers?

  7. This sum of fraction by denominator is called the totient function and is usually represented by the greek letter phi.

Additional Work

• • See this problem set from Julia Robinson Festival to take this further.