Need a "Prime Square" type of number (For my significant other)

Question

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.

Answer 1

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 :)

Answer 2

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).

Answer 3

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?