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Chapter 6: More advanced exercises

You will find in this tutorial a collection of more advanced exercises. You might also want to have a look at the worksheets on "Collatz conjecture", "Dictionaries and graph theory" and "Strings and the Burrows-Wheeler transform".

Exercise 6.1

How many strings of length n containing only letters a and b but not containing aa are there? Compute the value for each n = 1, 2, ..., 10

Do you know this sequence?

Could you find the corresponding entry in the OEIS

Exercise 6.2 (Goldbach conjecture)

Verify experimentally the following statement: « For each even integer n ≥ 4, there exist two prime number p ≥ 2 and q ≥ 2 such that n = p + q ».

Up to which value of n are you able to verify this conjecture?

Exercise 6.3

Does there exist two positive integer numbers x and y such that x2 − 61y2 = 1 ?

Exercise 6.4

Write a function that given positive integers (p, q, n) return the number of solutions in positive integers (a1, a2, …, an) to the equation

(1)/(a1) + (1)/(a2) + … + (1)/(an) = (p)/(q)

How many solutions are there for (p, q, n) = (13, 12, 3)?

A sample of Euler problems

Solve the following Euler problems

  • problem 26 (decimal expansions)
  • problem 31 (coin sums)
  • problem 45 (triangular, pentagonal and hexagonal numbers)
  • problem 46 (odd composite number that can not be written as the sum of a prime and twice a square)
  • problem 50 (sum of consecutive primes that are primes)

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