NTIC Exercises

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Exercises 13.7 Exercises

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1.

Prove that if $n\equiv 3\text{ (mod }4)\text{,}$ then $n$ cannot be written as a sum of two squares (13.1.1).

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2.

Prove Fact 13.1.2.

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3.

Show that if $n\equiv 7\text{ (mod }8\text{)}\text{,}$ then $n$ cannot be written as a sum of three perfect squares. (See also Exercise 14.4.6.)

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4.

Find two numbers that can be written as a sum of three squares in two essentially different ways (not just $1^2+0^2+0^2=0^2+1^2+0^2$ or even $3^2 + 4^2 +1^2 = 0^2 + 5^2 + 1^2$ ). (See also Exercise 14.4.4.)

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5.

Find as many integers $n$ as possible which are only writeable as a sum of squares via $n=a^2+a^2=2a^2\text{,}$ i.e. $n$ is not writeable as a sum of distinct squares.

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6.

Verify Fact 13.1.7 by hand (i.e. write all the algebra out).

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7.

Let $r_2(n)$ be the number of different ways to write $n$ as a sum of two squares, where every different way (not just essentially different) is counted. For instance,

$\begin{equation*} r_2(2)=4\text{ because }(-1,1),(-1,-1),(1,1),(1,-1)\text{ all work.} \end{equation*}$

Prove that

$\begin{equation*} r_2\left(2^m\right)=4\text{ for all }m\text{.} \end{equation*}$

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Let

$N$ be odd, and let

$N=a^2+b^2$ and

$N=c^2+d^2\text{,}$ where the pairs

$(a,b)$ and

$(c,d)$ are both positive and not the same or just switched in order. Verify the following to finish the proof of Fact 13.2.1.

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8.

It's okay to assume that $a$ and $c$ are odd and $b$ and $d$ are even, with $a\geq c$ and $d\geq b\text{.}$

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9.

If this is the case, show that $k=\gcd(a-c,d-b)$ and $n=\gcd(a+c,d+b)$ are both even.

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10.

Assuming the previous two exercises, show that $\frac{a-c}{k}=\frac{d+b}{n}$ and $\frac{d-b}{k}=\frac{a+c}{n}\text{.}$

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11.

Assuming everything else works, show that $N$ is in fact the product of the terms in question; this will involve a fair amount of cancellation!

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12.

Using the tools of this chapter, for each of the numbers $5095\text{,}$ $5096\text{,}$ $5097\text{,}$ $5098\text{,}$ and $5099\text{,}$ either write it as a sum of two perfect squares or explain why it is impossible to do so.

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Pick four random (to you) three digit numbers which are not of the form

$4k+3\text{.}$

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13.

Decide whether these numbers are a sum of two squares without using Sage.

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14.

Pick two of those numbers and write them in all possible ways as a sum of two squares.

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15.

Show a positive integer $k$ is the difference of two squares if and only if $k\not \equiv 2$ (mod $4$ ).

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16.

Prove that if $n\equiv 12$ (mod $16$ ), show that $n$ cannot be written as a sum of two squares.

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17.

Is there any congruence condition modulo $6$ for which a number cannot be written as a sum of two squares?

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18.

Referring to the proof of the main theorem (especially in Subsection 13.4.3): Check that the pictures you get from some other primes with these lattices really work.

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Check every piece of the Zagier proof (Proposition 13.6.1).

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19.

The set $S$ is finite. Try figuring out what $S$ is for $p=5$ or $p=13\text{,}$ the smallest such primes.

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20.

Each $(x,y,z)$ has exactly one of the three things to go to.

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21.

The function in question is an involution. That is, if you take the output and apply the function a second time, you get your original $(x,y,z)$ back (this is a little tougher).

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22.

If $(x,y,z)$ goes to $(x,y,z)$ then it turns out that $(x,y,z)=(1,1,\frac{p-1}{4})$ (you will probably need to use the definition of $S$ for this, and remember that we assume $p\equiv 1$ (mod $4)$ ).

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23.

That if the map $(x,y,z)\to (x,z,y)$ has a point which is fixed (the output is same as input) then this, combined with the definition of $S\text{,}$ means that $p$ is writeable as the sum of two squares.