prime-number
adjective
- Attributive form of prime number, noun.
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logic - A first order sentence such that the finite Spectrum of that sentence is the prime numbers - Mathematics Stack Exchange
Proper use of prime symbol and commas question.
The Prime Problem with a One Sentence Proof - Numberphile
The involution not shown is given by
(x + 2z, z, y - x - z) if x < y - z
(x,y,z) -> (2y - x, y, x - y + z) if y - z < x < 2y
(x - 2y, x - y + z, y) if x > 2y
which is verified by checking that, for example, (x + 2z)2 + 4z(y - x - z) = x2 + 4yz. (Note that y - z < 2y since x, y, z are defined to be positive.)
One can also check that it swaps the two cases x < y - z and x > 2y. Thus, to have a fixed point we must have y - z < x < 2y and we hence solve
(x, y, z) = (2y - x, y, x - y + z).
This is a system of linear equations whose general solution is x = y. So the question now becomes: when is x2 + 4xz = p? Since p is prime, we need x = 1 so this gives 1 + 4z = p and the fixed point is (1, 1, k) as Zagier claims.
Now to see that the number of solutions is odd (whence nonzero). We note that the involution above, gives an action of the 2-group, Z/2 on the set of solutions. By a corollary of the Orbit-Stabilizer Theorem, the set of fixed points and the set of solutions have the same residue modulo 2. Thus there are an odd number of solutions since the above involution has 1 fixed point.
More on reddit.comeli5:Can someone explain what a prime number is and WHY it’s prime number
Is 1 a prime number?
No, 1 is neither a prime number nor a composite number.
What makes a number prime?
If a number has only two factors (1 and itself), it is a prime number. If it has more than two factors, it is composite.
What is the lowest prime number?
The smallest prime number is 2.
Rather than giving an explicit answer, I'm going to give a hint. I saw this problem when I was in grad school, and I assume many other people did too. The secret to these problems is to have a huge library of mathematical results to draw on. Then you make up your answer to exploit some theorem that you already know. In this case, one way to start is to make a formula which forces the model to resemble an initial segment $\{1, \ldots, n\}$ of the natural numbers with relations for the restrictions of the graphs of the addition and multiplication functions to triples from that subset.
This exists due to very general results, namely that the set of primes is rudimentary. See this excellent survey on spectra: Durand et al. Fifty Years of the Spectrum Problem: Survey and New Results. The same holds true for all known "natural" number theoretic functions. Indeed, the authors remark in section 4.2 that "we are not aware of a natural number-theoretic function that is provably not rudimentary".
I'm having a little debate with my manager. I've been searching around and can't find an answer. We have a sentence that states:
We can handle handle your special printing needs for sizes up to 44" x 50,' in black and white or full color.
Note that the apostrophe (or, as it is referred to when noting sizes, the prime symbol), is outside of the comma. Is this correct?
I was under the impression that the prime symbol should be inside the comma:
We can handle handle your special printing needs for sizes up to 44" x 50', in black and white or full color.
Any thoughts?