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December 8

For each positive integer $n$, which primes $p$ are still primes in the ring $Z[e^{\frac{\pi i}{n}}]$?

For each positive integer $n$, which primes $p$ are still primes in the ring $Z[e^{\frac{\pi i}{n}}]$? When $n=1$, $Z[e^{\frac{\pi i}{n}}]$ is the original integer ring, when $n=2$, $Z[e^{\frac{\pi i}{n}}]$ is the ring of Gaussian integers, when $n=3$, $Z[e^{\frac{\pi i}{n}}]$ is the ring of Eisenstein integers, and the primes in the Gaussian integers are the primes $p\equiv 3mod4$, and the primes in the Eisenstein integers are the primes $p\equiv 2mod3$, but how about larger $n$? 218.187.66.163 (talk) 04:50, 8 December 2024 (UTC)

A minuscule contribution: for $n=4,$ the natural Gaussian primes $3$ and $7$ are composite:

• $3=(1-\omega -{\omega }^2)(1+{\omega }^2+{\omega }^3).$
• $7=(2-\omega +2{\omega }^2)(2-2{\omega }^2-{\omega }^3).$

So $11$ is the least remaining candidate.  --Lambiam 09:00, 8 December 2024 (UTC)

It is actually easy to see that $7$ is composite, since $2$ is a perfect square:

$2=i(1-i{)}^2={\omega }^2(1-{\omega }^2{)}^2.$

Hence, writing $\sqrt{2}$ by abuse of notation for $\omega (1-{\omega }^2),$ we have:

$\begin{array}{cc}7& =(3+\sqrt{2})(3-\sqrt{2})\\ & =(2\sqrt{2}+1)(2\sqrt{2}-1)\\ & =(5+3\sqrt{2})(5-3\sqrt{2})\\ & =(4\sqrt{2}+5)(4\sqrt{2}-5).\end{array}$

More in general, any natural number that can be written in the form $|2a^2-b^2|,a,b\in \mathbb{N},$ is not prime in $\mathbb{Z}[e^{\pi i/4}].$ This also rules out the Gaussian primes $23,$ $31,$ $47,$ $71$ and $79.$  --Lambiam 11:50, 8 December 2024 (UTC)

So which primes $p$ are still primes in the ring $Z[e^{\frac{\pi i}{4}}]$? How about $Z[e^{\frac{\pi i}{5}}]$ and $Z[e^{\frac{\pi i}{6}}]$? 220.132.216.52 (talk) 06:32, 9 December 2024 (UTC)

As I wrote, this is only a minuscule contribution. We do not do research on command; in fact, we are actually not supposed to do any original research here.  --Lambiam 09:23, 9 December 2024 (UTC)

Moreover, $-2$ is also a perfect square. (As in the Gaussian integers, the additive inverse of a square is again a square.) So natural numbers of the form $2a^2+b^2$ are also composite. This further rules out $11,$ $19,$ $43,$ $59,$ $67$ and $83.$ A direct proof that, e.g., $11$ is composite: $11=(1+{\omega }^2+3{\omega }^3)(1-3\omega -{\omega }^2).$ There are no remaining candidates below $100$ and I can in fact not find any larger ones either. This raises the conjecture:

Every prime number can be written in one of the three forms $a^2+b^2,$ $2a^2+b^2$ and $|2a^2-b^2|.$

Is this a known theorem? If true, no number in $\mathbb{Z}[e^{\pi i/4}]$ is a natural prime. (Note that countless composite numbers cannot be written in any of these forms; to mention just a few: $15,1155,2491.$)  --Lambiam 11:46, 9 December 2024 (UTC)

I'll state things a little more generally, in the cyclotomic field $\mathbb{Q}[e^{2\pi i/n}]$. (Your n is twice mine.) A prime q factors as $q=(q_1\cdots q_r{)}^{e_r}$, where each $q_i$ is a prime ideal of the same degree $f$, which is the least positive integer such that $q^f\equiv 1(\mathrm{mod}n)$. (We have assumed that q does not divide n, because if it did, then it would ramify and not be prime. Also note that we have to use ideals, because the cyclotomic ring is not a UFD.) In particular, $q$ stays prime if and only if $q$ generates the group of units modulo $n$. When n is a power of two times an odd composite, the group of units is not cyclic, and so the answer is never. When n is a prime or twice a prime, the answer is when q is a primitive root mod n. If n is 4 times a power of two times a prime, the answer is never. Tito Omburo (talk) 11:08, 8 December 2024 (UTC)

For your $n$, $n=3$ and $n=6$ are the same, as well as $n=5$ and $n=10$, this is why I use $e^{\frac{\pi i}{n}}$ instead of $e^{\frac{2\pi i}{n}}$. 61.229.100.34 (talk) 20:58, 8 December 2024 (UTC)

Also, what is the class number of the cyclotomic field $Q[e^{\frac{\pi i}{n}}]$? Let $\gamma (n)$ be the class number of the cyclotomic field $Q[e^{\frac{\pi i}{n}}]$, I only know that:

• $\gamma (n)=1$ for $n=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,27,30,33,35,42,45$ (is there any other such $n$)?
• If $m$ divides $n$, then $\gamma (m)$ also divides $\gamma (n)$, thus we can let $\phi (n)=\prod _{d|n}\gamma (n{)}^{\mu (n/d)}$
• For prime $p$, $p$ divides $\phi (p)$ if and only if $p$ is Bernoulli irregular prime
• For prime $p$, $p$ divides $\phi (2p)$ if and only if $p$ is Euler irregular prime
• $\phi (n)=1$ for $n=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,27,30,33,35,42,45$ (is there any other such $n$)?
• $\phi (n)$ is prime for $n=23,26,28,32,36,37,38,39,40,43,46,49,51,53,54,63,64,66,69,75,81,96,105,118,128,...$ (are there infinitely many such $n$?)

Is there an algorithm to calculate $\gamma (n)$ quickly? 61.229.100.34 (talk) 21:14, 8 December 2024 (UTC)

Can we say anything special about every pair of functions f,g, satisfying f(g(x))=f(x) for every x?

Especially, is there an accepted term for such a pair?

Here are three simple examples, for two functions f,g, satisfying the above, and defined for every natural number:

Example #1:

f is constant.

Example #2:

f(x)=g(x), and is the smallest even number, not greater than x.

Example #3:

f(x)=1 if x is even, otherwise f(x)=2.

g(x)=x+2.

2A06:C701:746D:AE00:ACFC:490:74C3:660 (talk) 09:31, 8 December 2024 (UTC)

One way to consider such a pair is dynamically. If you consider the dynamical system $x\mapsto g(x)$, then the condition can be stated as "$f$ is constant on $g$-orbits". More precisely, let $D$ be the domain of $f,g$, which is also the codomain of $g$. Define an equivalence relation on $D$ by $x\sim y$ if $g^a(x)=g^b(y)$ for some positive integers $a,b$. Then $f$ is simply a function on the set of equivalence classes $D/\sim$ (=space of orbits). In ergodic theory, such a function $f$ is thought of as an "observable" or "function of state", being the mathematical analog of a thermodynamic observable such as temperature. Tito Omburo (talk) 11:52, 8 December 2024 (UTC)

After you've mentioned temprature, could you explain what are f,g, as far as temprature is concerned? Additionally, could you give another useful example from physics for such a pair of functions? 2A06:C701:746D:AE00:ACFC:490:74C3:660 (talk) 19:49, 8 December 2024 (UTC)

This equation is just the definition of function g. For instance if function f has the inverse function f⁻¹ then we have g(x)=x. Ruslik_Zero 20:23, 8 December 2024 (UTC)

If f is the temperature, and g is the evolution of an ensemble of particles in thermal equilibrium (taken at a single time, say one second later), then because temperature is a function of state, one has $f(x)=f(g(x))$ for all ensembles x. Another example from physics is when $g$ is a Hamiltonian evolution. Then the functions $f$ with this property (subject to smoothness) are those that (Poisson) commute with the Hamiltonian, i.e. "constants of the motion". Tito Omburo (talk) 20:33, 8 December 2024 (UTC)

Thx. 2A06:C701:746D:AE00:ACFC:490:74C3:660 (talk) 10:43, 9 December 2024 (UTC)

Let $f$ be a function from $X$ to $Y$ and $g$ a function from $X$ to $X.$ Using the notation for function composition, the property under discussion can concisely be expressed as $f\circ g=f.$ An equivalent but verbose way of saying the same is that the preimage of any set $B\subseteq Y$ under $f$ is closed under the application of $g.$  --Lambiam 08:54, 9 December 2024 (UTC)

Thx. 2A06:C701:746D:AE00:ACFC:490:74C3:660 (talk) 10:43, 9 December 2024 (UTC)

IEEE Xplore paper claim to acheive exponentiation inversion suitable for pairing in polynomial time. Is it untrustworthy ?

I just found https://ieeexplore.ieee.org/abstract/document/6530387. Given the multiplicative group factorization in the underlying finite field of a target bn curve, they claim to acheive exponentiation inversion suitable for pairing inversion in seconds on a 32 bits cpu.

On 1 side, the paper is supposed to be peer reviewed by the iee Xplore journal and they give examples on 100 bits. On the other side, in addition to the claim, their algorithm 2 and 3 are very implicit, and as an untrained student, I fail to understand how to implement them, though I fail to understand things like performing a Weil descent.

Is the paper untrustworthy, or would it be possible to get code that can be run ? 2A01:E0A:401:A7C0:152B:F56C:F8A8:D203 (talk) 18:53, 8 December 2024 (UTC)

About the paper, I agree to share the paper privately 2A01:E0A:401:A7C0:152B:F56C:F8A8:D203 (talk) 18:54, 8 December 2024 (UTC)