Section3.6Exercises

  1. For each of the following linear Diophantine equations, either find the form of a general solution, or show there are no integral solutions.
    • \(21x+14y=147\)
    • \(30x+47y=-11\)
  2. Find all integer solutions to the following system of equations. (Hint: do what you would ordinarily do in high school algebra or linear algebra! Then finish the solution as we have done.)
    • \(x+y+z=100\)
    • \(x+8y+50z=156\)
  3. Explore the patterns in the positive integer solutions to \(ax+by=c\) situation above. For sure I want you to do this for the ones I mention there, but try some others and see if you see any broader patterns!
  4. Give me as much information about the conductor as you have thus far accumulated.
  5. Compute the number of positive solutions to the linear Diophantine equation \(6x+9y=c\) for various values of \(c\) and compare to the analysis we did above.
  6. Prove that any line \(ax+by=c\) which hits the integer lattice but \(gcd(a,b)\neq 1\) is the same as a line \(a'x+b'y=c'\) for which \(gcd(a',b')=1\), and explain why that means that without loss of generality the first topic doesn't need any more explanations.
  7. Find a primitive Pythagorean triple with at least three digits for each side.
  8. Prove that 360 cannot be the area of a primitive Pythagorean triple triangle. Is it be the area of some Pythagorean triple triangle? (See optional section.)
  9. Find a way to prove that \(x^4+y^4=z^4\) is not possible for any three positive integers \(x,y,z\).
  10. We already saw that if \(x,y,z\) is a primitive Pythagorean triple, then exactly one of \(x,y\) is even (divisible by 2). Assume that it's \(y\), and then prove that \(y\) is divisible by 4.
  11. Under the same assumptions as in the previous problem, prove that exactly one of \(x,y,z\) is divisible by 3. (This proves that every area of a Pythagorean triple triangle is divisible by 6. It is also true that exactly one of \(x,y,z\) is divisible by 5.)
  12. Write down your own definition of a prime number. Then compare it with the book, a few internet sources, etc. Should 1 be considered prime? What about \(-1\)?
  13. Search books and/or the Internet and find at least three different proofs that there is no largest prime number. You don't have to understand all the details; they should be fairly different from each other, though. Do any of the proofs generate all primes in order?
  14. Bonus: Prove this fact about gcds \[\gcd(x,y)^2=\gcd(x^2,xy,y^2)\] Hint: use \(\gcd(a,b,c)=\gcd(\gcd(a,b),c)\) and try to somehow factor out \(\gcd(a,b)\) cleverly, or use the definition of gcd as the biggest of some set of divisors.
  15. Find a (fairly) obvious solution to the equation \(m^n=n^m\) for \(m\neq n\). Are there other such solutions?
  16. Prove that 360 cannot be the area of a primitive Pythagorean triple triangle.
  17. Prove that in a primitive Pythagorean triple \(x,y,z\), if \(y\) is the even one, it is divisible by 4, and that exactly one of \(x,y\) are divisible by 3.