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# Math 404/504 homework

## Homework 2, due Thursday 27 January

- Show that if $u,v$ have no common factors, aren’t both even, and aren’t both odd, then the triple $(u^2-v^2,2uv,u^2+v^2)$ will have no common factors.
- Silverman, exercise 3.2
- Show that if $r_i,r_{i+1},r_{i+2}$ are successive remainders in a run of the Euclidean algorithm, then $r_{i+2}\lt\frac12r_i$. (See Silverman, exercise 5.3.)
- Let $F_n$ denote the Fibonacci sequence. Show that in a run of the Euclidean algorithm on $F_{n+1},F_n$, the quotient $q$ will be $1$ every time. Show that the algorithm terminates in $n-1$ steps.

### Supplemental problems

- SP5 Silverman, exercise 3.5(a)-(d)
- SP6 Find the $\gcd$ of any pair of Fibonacci numbers $F_n$ and $F_m$ in terms of $n$ and m$, and prove you are correct.
- SP7 Silverman, exercise 5.4

## Homework 1, due Thursday 20 January

- Show how to generate a square triangle number from a pair $p,q$ such that $p^2$ and $2q^2$ are one unit apart. Verify your resulting quantity $n$ is a square triangle.
- Show how, given $p,q$ as above, to generate a new pair $p’,q’$ with the same property. Verify your $p’,q’$ satisfy that $(p’)^2$ and $2(q’)^2$ are one unit apart.
- Show that if $(a,b,c)$ is a primitive Pythagorean triple, then exactly one of $a,b$ is odd.
- Show how to generate a primitive Pythagorean triple from two odd numbers $s,t$. Verify that it is a Pythagorean triple.
- Silverman, exercise 1.6.

### Supplemental problems

- SP1 Silverman, exercise 1.3
- SP2 Silverman, exercise 2.1
- SP3 Silverman, exercise 2.6
- SP4 Silverman, exercise 2.7
- … suggest your own!