NTIC Exercises

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Exercises 13.7 Exercises

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1.

Prove that if $n \equiv 3 (mod 4),$ 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 $n \equiv 7 (mod 8),$ 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 $n = a^{2} + a^{2} = 2 a^{2},$ 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 $n \geq 0$ as a sum of two squares, where every different way (not just essentially different) is counted. For instance,

r_{2} (2) = 4 because (- 1, 1), (- 1, - 1), (1, 1), (1, - 1) all work.

Prove that

r_{2} (2^{m}) = 4 for all m \geq 1 .

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Let $N$ be odd, and let $N = a^{2} + b^{2}$ and $N = c^{2} + d^{2},$ 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 $d \geq b .$

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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 $\frac{d - b}{k} = \frac{a + c}{n} .$

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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 $5095,$ $5096,$ $5097,$ $5098,$ and $5099,$ 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 $4 k + 3 .$

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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 $k ≢ 2$ (mod $4$ ).

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16.

Prove that if $n \equiv 12$ (mod $16$ ), then $n$ 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 $p = 13,$ 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 $S,$ means that $p$ is writeable as the sum of two squares.

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24.

Prove the assertion about $\pm (\frac{p - 1}{2})!$ in Remark 13.3.4.