This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?
PS: feel free to interpret the term natural in a broad sense; I only included it to avoid answers along the lines of "take [fact about the primes] $\to$ [string of connections between various areas of mathematics] $\to$ [geometry!]"
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Well, prime numbers are strongly related to the Riemann zeta function, $\zeta(s)$. This has a product representation which involves the roots of the function. The Riemann Hypothesis now states that all non-trivial roots in the complex plane lie on the "critical line": $$Re(z)=\frac12$$ which can be thought of as a geometric feature.
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Here’s an example, far from the best, of prime numbers entering into a (relatively) geometric problem. Consider all the points on the unit circle, $X^2+Y^2=1$. Notice that by considering this as the set of complex numbers $a+bi$ of absolute value one, i.e. $a^2+b^2=1$, this has a natural group structure. Explicitly, $(a,b)*(c,d)=(ac-bd,ad+bc)$.
Now, here’s the question: What are the rational points on the circle? That is, what are the points $(a,b)$ on the circle for which both $a$ and $b$ are rational numbers? Your first interesting case is $(3/5,4/5)$. Of course there’s an answer to this question coming from the classical solution to the problem of finding all Pythagorean Triples. But I want to ask an arithmetic question: What are the possible denominators of all the rational points on the circle?
The answer comes out of looking at the “primes” in the ring of Gaussian Integers, but I’ll cut to the chase: a number will appear as the (common) denominator $D$ of a rational pair $(a,b)$ on the unit circle if and only if the only primes dividing $D$ are those of the form $4k+1$. Naturally, I want the rational numbers $a$ and $b$ to be in lowest terms.
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A simple example: number p>2 is prime iff any equiangular p-gon with rational side lengths is regular, see, e.g., http://www.cut-the-knot.org/Outline/Geometry/EquiangularP-gon.shtml
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Events on the horizon of a 2D Universe
You may have a look to the graph; it represents all the numbers (red, up to 100) that "see" straight to the origin. e.g. prime 7 is represented by the seventh vertical point-column, prime 13 by the 13th column. The uniqueness of prime numbers is obvious.
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Construction of polygons that have prime number properties are more complex to construct than composite numbers, a polygon with composite properties can easily constructed by splitting each compostite fraction into half, example, circle in half = 2 Circle in quarters = 4 Continue Quarters in half = 8 (notice 6 is missing in this sequance) because a circle is devided into six by its own radius example would be sacred geometry (creating flower of life)
Here is my layout of prime numbers... 01 03 05 07 11 13 17 19... 02 06 10 14... 04 09 15... 08 12 20... 16 18...
Notice the 1st numbers going across are prime Notice the 1st numbers going down are composite and are easy to use when constructin pygons (polygons related to geometry) search polygon for further understanding.
Iv noticed prime numbers are never a even number, though i have not studied further on this, only up to 20 that i studied.
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How about another imagery about prime numbers? A prime number $p = 1 \times p$ and hence, geometrically it is like a one-dimensional segment. On the other hand, a composite number $c= a \times b$, where $a$ and $b$ are its prime factors, is like a rectangle having an area $c$, with side lengths $a$ and $b$ . So, in general, composite numbers $c$ may be imagined as multi-dimensional rectangular parallelepipeds with volumes $c= a \times b \times c \times d \cdots$, having side lengths corresponding to their prime factors. Of course, the question that is pertinent is: does this imagery lead to interesting results or insights?
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Let there be an infinite triangular lattice. Sequentially enumerate the triangles in a constant direction and a 6 cell hexagon forms. However, if one enumerates the cells such that each time a prime number is encountered the direction of fill switches, the number line exits the 6 cell, but forms a new hexagon (24 cells) - confining all natural numbers to infinity. The reason for confinement is all primes (aside from 2 and 3)are 1Mod6 or 5Mod6 and get stuck in the middle of the 24 cell. If you are interested in more details, see my woefully outdated website: hexspin.com
The Gauss-Wantzel theorem on constructible polygons immediately springs to mind. This states that a regular $n$-gon is constructible with a straightedge and compass iff $n$ is the product of a power of $2$ and a collection of distinct Fermat primes.
The power of $2$ is only there because if you can construct an $n$-gon, you can easily construct a $2n$-gon by constructing an isoceles triangle on each side of the $n$-gon. Doing this repeatedly, you can get a $2^mn$-gon. So really, this is about the nature of Fermat primes.