# Tag Info

### What actually is the idea behind the condensed mathematics?

I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written ...

### Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?

The discriminants of the irreducible polynomials $$(x^2-2)^2+60 = x^4 - 4 x^2 + 64, \qquad (x^2+2)^2+60 = x^4 + 4 x^2 + 64$$ are both equal to $58982400 = 2^{18} \cdot 3^2 \cdot 5^2$. However, the ...

### A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent

The equation (1) for a fixed $x$ is equivalent to the congruence: $$y^2 \equiv 2(1+z^2)\pmod{x}.$$ For $x=25964951$, we have $2\equiv 3328351^2\pmod{x}$, and thus all solutions are obtained from those ...

### Priming for the primes

This concerns prime powers and not strictly primes, but the fact that the topological Tverberg conjecture (see e.g. https://arxiv.org/abs/1605.05141) holds for prime power values of “r” and not any ...

### Priming for the primes

Virtually all important examples of error-correcting codes that have a chance to be optimal and practical in applications utilize prime numbers.
1 vote

### Priming for the primes

Prime numbers occur in a major way in the theory of tactical configurations, in particular in finite geometries.

### class number and negative Pell equation

Added: my indefinite forms are reduced in the sense of Gauss and Lagrange. That is, $\langle a, b, c \rangle$ means the form $f(x,y) = a x^2 + b xy + c y^2$ with discriminant $\Delta = b^2 - 4ac.$ ...

### Priming for the primes

Feedback with carry shift register sequences (FCSRs) are an arithmetic parallel of the LFSRs in the answer by @WlodAA. They can be represented using $N-$adic numbers and achieve maximal period when $N$...

### Priming for the primes

Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information ...

### Priming for the primes

For me, the most spectacular fact that uses primes in an essential way and which no prime number theorist would be interested in is the fact that Tarski monsters exist for all sufficiently large ...

### Priming for the primes

(Very large) prime numbers are of particular importance in cryptography. See e.g. RSA algorithm. I remember also some quasi random number generators using prime numbers.

### Priming for the primes

Primes even appear outside mathematics, here is an example from biology: The fact that some species of cicadas appear every 7, 13, or 17 years and that these periods are prime numbers has been ...

### Priming for the primes

Regular polygons which you can construct with compass and straightedge have $2^k \cdot p_1 \cdot p_2 \dots p_n$ edges, where $p_i$ are all distinct Fermat primes (so not all primes). It's the reason ...

### Priming for the primes

There are maximal shift registers of length $\ p^k-1\$ where $\ p\$ is an arbitrary prime, and $\ k\$ is an arbitrary natural number.

### Priming for the primes

Obvious, but still worth mentionning, because the trick is frequently helpful: to represent a collection of elements of different types, where you can have more than one element per type, and order ...

### Priming for the primes

The nonzero characteristics of fields are precisely the prime numbers.

### Priming for the primes

The abelian simple groups are precisely the groups of prime order.

### Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices

The approach indicated by @AlexanderKalmynin is a good heuristic, for sure. Still, there is a somewhat larger context that may be more explanatory, depending on one's tastes. Namely, we can see that ...

### Computing Thompson series for the monster group

In the OEIS index, see "McKay-Thompson sequences or series for Monster simple group, sequences related to" These may come with tables of coefficients, and with programs to compute ...

### A question about generalized harmonic numbers modulo $p$

Glaisher's I-numbers are described in J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
1 vote

### Understanding inequality in "Small gaps between primes"

For your questions it suffices to study the inner sum $$\sum_{\substack{u_1,....,u_k < R \\ p\mid u_i,u_j \\ (u_i,W)=1 \forall i}}\prod_{i=1}^k \frac{\mu(u_i)}{\varphi(u_i)}.$$ As was pointed out ...

### The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

FWIW, I found a source according to which He found this by first squaring the square of π which gives 97.40909103…, then by subtracting 9^2 = 81 he got 16.40909103…, multiplying this by 22 he got 361....
Accepted

I will answer Q2: N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that $Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12]... 5 votes Accepted ### Computing mth power residue symbols I am not sure what you mean by "to resort to these symbols". If you want to compute their values given$\alpha$and${\mathfrak b}$, just use the definition. Explicit reciprocity laws for ... 2 votes ### Reference for universal elliptic curves To complete Lennart Meier's nice answer, Baaziz has computed explicit equations for the universal elliptic curve over$Y_1(N)$up to$N=51$. The method used and the data up to$N=20$can be found in ... 4 votes Accepted ### Reference for universal elliptic curves For any$n\geq 1$, one can define a functor$\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a ... 4 votes Accepted ### Bounds for the sequence$a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$Let us show that \begin{equation*} a(n)\le\exp(n^{1-A+o(1)}) \tag{1}\label{1} \end{equation*} (as$n\to\infty$). Indeed, for each$q\in(1-A,1)$, \begin{equation*} (k-1)^q-k^q\sim-qk^{q-1},\... 1 vote ### Representations of$\zeta(3)$as continued fractions involving cubic polynomials In a comment to a related question, an article from 1980 (!) is quoted, where this has been proved as a by-product on p. 20 in a quite elementary way. Thus a by-product indeed, but not by Ramanujan ... 2 votes Accepted ### Bounds on Bézout coefficients We may suppose that$-m<k\leqslant 0$and choose integers$y_i$such that$\sum y_ia_i=k$. Next, by replacing$(y_1,y_i)\to (y_1\pm a_i, y_i\mp a_1)$we may achieve$y_i\in [0,a_1)$for all$i>1$... 11 votes Accepted ### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist? I will show the two results are non-superficially related by showing one of them implies the other: the classification of moduli$n \geq 2$for which the unit group$(\mathbf Z/(n))^\times$is cyclic ... 17 votes ### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist? Both Gauss' generalization, and the classification of moduli with primitive roots, are 'shadows' of the structural theory of the finite abelian group$G_m:=(\mathbb{Z}/m\mathbb{Z})^{\times}$. Gauss' ... 2 votes Accepted ### Overconvergent modular forms and the level at$p$The curve$X_1(Np^n)$is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union ... 1 vote ### Bounds on Bézout coefficients The following is likely useful, but doesn't answer your question really. First, a standard (set of) inequalities used to bound the covering radius of a lattice is known by the name of transference. ... 2 votes ### On the Hilbert function of a numerical semigroup I believe there is an easier proof. Proof. By induction, it suffices to prove the case that$eS^* = e + (e - 1) S^*$: indeed, if we assume that$(n + 1) S^* = e + nS^*$for some integer$n \geq e,$... 2 votes ### Twin Primes- Clement conjecture proof It follows immediately (mechanically!) by$\rm\color{#90f}W$= Wilson's theorem and (Easy) CRT as below, using$\!\bmod n\!+\!2\!:\ \color{#0a0}{(n\!+\!1)!} = \smash{\underbrace{(n\!+\!1)n}_{\large \ \...
Yes, the series diverges. We can reduce easily to the case of irreducible monic $Q$. Next, let $\alpha_Q(p)$ be the number of roots of $Q(x)$ in $\mathbb{Z}/p\mathbb{Z}$. Note that $M_Q$ is the set of ...