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    "$$\n",
    "\\def\\CC{\\bf C}\n",
    "\\def\\QQ{\\bf Q}\n",
    "\\def\\RR{\\bf R}\n",
    "\\def\\ZZ{\\bf Z}\n",
    "\\def\\NN{\\bf N}\n",
    "$$\n",
    "# Chapter 6: More advanced exercises\n",
    "\n",
    "You will find in this tutorial a collection of more advanced exercises.\n",
    "You might also want to have a look at the worksheets on \"Collatz\n",
    "conjecture\", \"Dictionaries and graph theory\" and \"Strings and the\n",
    "Burrows-Wheeler transform\".\n",
    "\n",
    "## Exercise 6.1\n",
    "\n",
    "Does there exist two positive integer numbers $x$ and $y$ such that\n",
    "$x^2 -\n",
    "61y^2 = 1$ ?"
   ]
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   "source": [
    "## Exercise 6.2\n",
    "\n",
    "How many strings of length $n$ containing only letters `a` and `b` but\n",
    "not containing `aa` are there? Compute the value for each $n=1,2,...,10$"
   ]
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    "# edit here"
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   "source": [
    "Do you know this sequence?"
   ]
  },
  {
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   "execution_count": null,
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   "outputs": [],
   "source": [
    "# edit here"
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   "source": [
    "Could you find the corresponding entry in the [OEIS](https://oeis.org/)"
   ]
  },
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   "source": [
    "## Exercise 6.3 (Goldbach conjecture)\n",
    "\n",
    "Verify experimentally the following statement: \u00ab For each even integer\n",
    "$n \\geq\n",
    "4$, there exist two prime number $p \\geq 2$ and $q \\geq 2$ such that\n",
    "$n = p +\n",
    "q$ \u00bb."
   ]
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   "source": [
    "Up to which value of $n$ are you able to verify this conjecture?"
   ]
  },
  {
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    "# edit here"
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   "source": [
    "### More\n",
    "\n",
    "Solve the following Euler problems\n",
    "\n",
    "-   [problem 26](https://projecteuler.net/problem=26) (decimal\n",
    "    expansions)\n",
    "-   [problem 31](https://projecteuler.net/problem=31) (coin sums)\n",
    "-   [problem 45](https://projecteuler.net/problem=45) (triangular,\n",
    "    pentagonal and hexagonal numbers)\n",
    "-   [problem 46](https://projecteuler.net/problem=46) (odd composite\n",
    "    number that can not be written as the sum of a prime and twice a\n",
    "    square)\n",
    "-   [problem 50](https://projecteuler.net/problem=50) (sum of\n",
    "    consecutive primes that are primes)"
   ]
  }
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