Homework 10

This assignment is due by the beginning of class on Monday, April 27th. Be sure to review the homework guidelines before getting started.

  • Given a primitive Pythagorean triple, x, y, and z, prove that the hypotenuse z is congruent to 1 mod 4.
  • Find all sibling triples for hypotenuse of 85 using the Fibonacci Identity. Be sure to include the number of sibling triples in your answer.
  • The first few Germain primes are 2, 3, 5, 11, 23 . Find the next two.
  • Find a Germain prime p so that 2p+1 \nmid 2^p-1. Does this contradict the theorem stated in class?
  • Fermat's Last Theorem states that the equation a^n + b^n = c^n has no positive integral solutions a,b,c with n > 2. Prove the special case: p^3 = x^3 + y^3 has no solutions in positive integers x,y and p is prime.
  • Express \frac{120}{43} as a finite simple continued fraction.
  • What number is [\overline{1,2}]?
  • Compute the first 5 convergents of [\overline{1,2}]. Find the best rational approximation to [\overline{1,2}] with denominator at most 30.
  • Can you give a proof that Goldbach's (strong) conjecture implies Levy's (medium) conjecture? If so, do it. If not, why not?
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