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3 votes

Infinitely many primes that split completely in an arithmetic progression

Let $L$ be a Galois extension whose discriminant is relatively prime to the discriminant of $K$. Then $K\cap L = \mathbf Q$ since their discriminants are relatively prime: a prime that ramifies in $K ...
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2 votes

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

Another counterexample: $f_1(x)=x^3+9x+20, f_2(x)=x^3+6x+18$, discriminant = $4*27*73$ for both $f_1$ and $f_2$. Both polynomials are irreducible over $\mathbb{Q}$ by the rational root test. The 2-...
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2 votes

Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$

A partial answer: for fixed $m$, I provide a formula to compute a polynomial expression for $D_m(n)$ valid for all $n \geq m+1$. I'm not immediately sure if the expression will still be valid for $n &...
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31 votes

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 ...
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16 votes

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 ...
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2 votes

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 ...
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2 votes

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 ...
2 votes

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.
2 votes

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 $\Delta = b^2 - 4ac.$ ...
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2 votes

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 $N$...
7 votes

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 ...
4 votes

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 ...
5 votes

A question involving the three-dimensional Kloosterman sum

The proof 1 of Remark 6.2 in Lectures on applied $\ell$-adic cohomology works without modification for an arbitrary rational linear transformation and thus applies to $$\sideset{_{}^{}}{^{\ast}_{}}\...
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3 votes

Priming for the primes

The prime numbers come in handy when studying countability. For example, one can prove that the set of finite subsets of $\mathbb{N}=\{1, 2, 3, \ldots\}$ is countable basically by considering the ...
3 votes

Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?

I like the approach of looking for a sequence of integers with prime divisors that are guaranteed to be in a certain arithmetic progression. One can try taking the iterates of a polynomial, starting ...
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1 vote

Question on consecutive integers with similar prime factorizations

This is how Heath-Brown resolved the d(n)=d(n+1) problem ! The idea is due to Claudia Spiro who used it for d(n)=d(n+5040) but was unable to reduce the 5040. The same idea has now been used in many ...
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7 votes
Accepted

Small covering of divisors

UPDATED We can simply take $$B = \{1\} \cup \{ d\in D_n\ :\ d > n^{1/2} \}.$$ Then for any $d\in D_n$: if $d\leq n^{1/2}$, we take $(a,b)=(d,1)$; if $d>n^{1/2}$, we take $(a,b)=(1,d)$. Then $$\...
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2 votes

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.
11 votes

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 ...
6 votes

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 ...
3 votes

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.
6 votes

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 ...
15 votes

Priming for the primes

The nonzero characteristics of fields are precisely the prime numbers.
12 votes

Priming for the primes

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

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 ...
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0 votes

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

If $h'=h$, then for each $h''\in L^*$ one has $\exp(2\pi iQ(h-h',h''))=1$, so your sum is equal to $$ |\det Q|^{-1} |L^*/L|=1. $$ On the other hand, if $h\neq h'$, then there is an element $h_0\in L^*/...
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11 votes

Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

The (fairly poetic and ill-formed) idea in this story is that the Kapustin-Witten story and the Langlands program are about the SAME four-dimensional TQFTs, but evaluated on different "manifolds&...
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10 votes
Accepted

Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

A curve $C$ over $\mathbb F_q$ has dimension $3$ in this perspective (which is why you get a vector space) and a local field has dimension $2$ (which is why you get a category. So one only has to go ...
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1 vote

Localization at multivariate monic polynomials

Here is a strengthening of the result established by Will Sawin: Claim 1. Let $R$ be a commutative ring with identity. Let $n$ and $k$ be positive integers and let $\prec$ be a monomial order on $R[...
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2 votes

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 ...
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  • 37.2k
2 votes

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.
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  • 13.3k
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 ...
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2 votes

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....
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2 votes
Accepted

Non-vanishing modular forms

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]...
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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 ...
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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 ...
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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 ...
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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},\...
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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 ...
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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 $i>1$...
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  • 88.5k
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 ...
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  • 41.8k
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' ...
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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 ...
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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. ...
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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 $n \geq e,$ ...
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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 \ \...
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  • 4,578
9 votes
Accepted

Divergence of primes dividing polynomials

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 ...
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1 vote

Reverse engineering a Diophantine equation

It can't be done. By the MRDP Theorem, diophantine sets are the same as c.e. (or Turing Recognizable) sets, and therefore the problem of determining whether two diophantine equations $\varphi, \hat\...
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1 vote

On the Hilbert function of a numerical semigroup

Let $\text{e}$ be the minimum of $S^*$. Call a height function a function from $\mathbb Z/\text{e} \mathbb Z$ to the natural numbers. The height function $A$ for the numerical semigroup $S^*$ is ...
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