Homework 10

This assignment is due by the beginning of class on Monday, April 27^{th}. 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?