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No this isn't for a silly math exercise, it's a relationship with a hottie I don't want to lose at stake:

I like my TV volume to be on Perfect Squares, but she likes her volumes on Prime Numbers.

Easily provable is that for $n \geq 2$, there's no such number. $n = 0$ or $1$ are not prime numbers.

Mute is our current compromise, but for some programs that's just not feasible. If you're going to say "You're crazy/she's crazy", save it: I know.

I'm willing to stretch the rules to make this work. For example, my friends surround sound is actually measured in -dB, in which case -9 db would be a perfect volume (0 + 3i), both parts of that complex number satisfy one of us.

Is there any work arounds you guys can think of? Any crazy sketchy proof of why there exists a number (preferably positive, I dont wanna buy a new speaker system) that works for us?

Please Mathematics. This girl is out of my league beautiful, and into me.

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Are you asking why there are infinitely many $(a+bi)$ where $\pm (a+bi)^2$ is a square and $a$ or $b$ is prime? Just take Pythagorean triples.. –  Cocopuffs Jun 22 '12 at 17:45
    
I was hoping somebody would have a clever answer that didn't involve me having to go get a new stereo –  SetSlapShot Jun 22 '12 at 17:49
    
Why is it ok to consider numbers of the form $p i$ to be "prime" for $p$ prime? –  fretty Jun 22 '12 at 18:59
    
Make some research on amicable numbers and twin primes then! –  Pedro Tamaroff Jun 22 '12 at 21:26

3 Answers 3

up vote 5 down vote accepted

So here's a good choice: 17.

First of all, it is a prime, so she gets what she wants. And it's only 1 away from being a square, so you almost get what you want. But wait, there's more.

17 = 16+1, the sum of 2 squares

17 = 4+4+9, the sum of 3 squares

17 = 4+4+4+4+1, the sum of 5 squares

(with 2,3, and 5 being the first three primes)

17 = 17, the sum of 1 prime

17 = 2+3+5+7, the sum of 4 primes (also consecutive)

(with 1 and 4 being the first two squares)

17 a cousin prime (differs from 13 by 4) and, more important to your cause, a doubly sexy prime (11, 23).

Furthermore, 1/17 written as a decimal repeats every 16 digits, which is not only square, it's also the same square as the $n^2+1$ formulation.

I hope this can help you with your socio-audio dilemma :)

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+1. $17$ also decomposes as $1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$, another sum of a prime number of squares. –  Blue Jun 22 '12 at 21:21
    
Haha, I guess you're right. –  Eric Stucky Jun 23 '12 at 1:20
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You are a champion. I had never thought of 17 as so friendly to me...she said that's always been her lucky number! It was meant to be. Thank you sir. –  SetSlapShot Jun 25 '12 at 12:40

I think you should compromise, and settle on a number which is comfortably nearly prime, and/or nearly a square.

You might argue for the square of a prime (such as 4, 9, 25, ...) since those only have one prime divisor (so you get what you want, and she "nearly" does).

Or you could argue for the product of consecutive primes (such as 6, 15, 35, 77, ...) as these are in a sense "nearly" prime and "nearly" square (so neither of you get what you want).

Or, you might argue for a prime (so this gives her what she wants) of the form $n^2+1$ (such as 2, 5, 17, 37, 101, ...) since those differ from a square by as little as possible (it is conjectured, but unproven, that there are infinitely many such primes to choose from).

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I do see myself willing to comprimise and make it work for what is conjectured to be the smallest distance from a prime. Thank you. I'll make sure your invitation to the wedding doesn't get lost in the mail. –  SetSlapShot Jun 22 '12 at 19:12

Maths is not a tool invented in order to help you "get a girl".

If you are suddenly willing to accept things like $-9 = (3i)^2$ purely on the basis that $3$ is prime then why couldn't you do this before?

We have that 9 = 3^2 and $3$ is also a "complex number" containing a prime. Why is this a problem when your new rules allow it?

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Because it's 0 + 3i**2. I don't get my jollys off from the 3, but the 0 (a perfect square) –  SetSlapShot Jun 22 '12 at 19:13
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"Maths is not a tool invented in order to help you get a girl". The unreasonable effectiveness of mathematics strikes again! –  Cam McLeman Jun 22 '12 at 19:14
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True, mathematics wasn't invented for that. That's what applied math is for. –  Matthew Conroy Jun 22 '12 at 19:17
    
Well I do not understand the problem then. You are allowing $-9$ because it is $(3i)^2$ and $3$ is prime yet you are not allowing $9 = 3^2$. You can't change the rule AND expect the same rule to hold in the previous case! –  fretty Jun 22 '12 at 19:25
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Maths is not a tool invented in order to help you "get a girl". I disagree with you on a fundamental level. Thank you for the input. –  SetSlapShot Jun 25 '12 at 12:41

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