共 303 个问题，第 15/16 页
Is there a research program that attempts to reconstruct an underlying structure from the statistical properties of the Riemann zeros?
I am a curious outsider to mathematics and recently started reading about the Riemann Hypothesis. I am aware that many outsiders mistakenly believe they have solved the Riemann Hypothesis. I am not making such a claim. I am only trying to understand whether this perspective already exists in the...
Does $x_1^n + x_2^n + \dots + x_n^n = z^n$ have infinitely many primitive solutions in positive integers?
I am familiar with "Fermat's Last Theorem" and the disproved "Euler's sum of powers conjecture". By my understanding, the latter conjecture states that $n$ terms are required to have solutions, which has been disproved by counterexample. My question is whether or not you can always find an...
Infinitude and asymptotic growth of a sparse recursively generated prime sequence
Construction of the set: Start with an empty set F.Test primes in sequential order.A prime P is called "foundational" and belongs to F if and only if it is NOT representable as $$x_1 q_1^{a_1}+x_2 q_2^{a_2}+...+x_n q_n^{a_n}$$ Where ${q_1 ,q_2, q_3,...,q_n}$ are earlier primes of the set F and...
Minimum value of $i$ to change $\lfloor n / i \rfloor$
I was solving this CSES question called Sum of Divisors and one of the solutions in the USACO Guide hints at this statement: The minimum value of $j > i$ such that $\lfloor n/j \rfloor$ < $\lfloor n/i \rfloor$ is $j = \lfloor n/q \rfloor + 1$, where $q = \lfloor n/i \rfloor$ for integers $j, i...
Finding the equation of a tangent line to a projective curve at a non-singular point.
I am currently working through Fulton's Algebraic Curves and I have attempted the following problem: $$\text{Let P be a simple (non-singular) point on }F\text{ . Show that the tangent line to }F \text{ at } P\text{ has the equation }F_X (P )X + F_Y (P )Y + F_Z (P )Z = 0$$ My solution thus far...
An elliptic curve for $x_1^5+x_2^5+x_3^5=y_1^5+2y_2^5$?
In a prior post, the equation, $$x_1^5+2x_2^5 = y_1^5+2y_2^5$$ was considered. It has only one known primitive solution. This present post considers the similar, $$x_1^k+x_2^k+x_3^k = y_1^k+2y_2^k$$ valid for both $k = (1,5)$. Duncan Moore found only one primitive solution, namely, $$85333^k +...
Direct calculation of $H_1$ of the cotangent complex
Let $k$ be a commutative ring and let $A$ be a commutative $k$-algebra. The cotangent complex $\mathbf{L}_{A \mid k}$ can be computed using a simplicial resolution of $A$ as follows: choose a simplicial commutative $k$-algebra $P$ and an augmentation $\epsilon : P_0 \to A$ (i.e. a $k$-algebra...
What conditions are needed for intersection number of Cartier divisors to equal dimension of global sections?
Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed)....
Number theory - why does the dot product on the Minkowski embedding resemble the Frobenius inner product?
The trace form $(a,b)\mapsto \mathrm{tr}(ab)$ is easily motivated as a choice of bilinear form on a number field $K$ by noting that it agrees with the (standard real) dot product on $\mathbb{R}^{r_1}×\mathbb{C}^{r_2}$ as restricted to the Minkowski embedding of $K$ (the one that sends a number...
Dyadic dissection with major arcs
In Vaughan's paper "A variance for k-free numbers in arithmetic progressions" https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024611505015352 he uses at one point a kind of dyadic decomposition for major arcs. From my understanding, I think he has $$\sum _{q\leq R}\int_{|\beta...
Computing Cartier Divisor from Weil Divisor Example
I am trying to work through the following problem: Let $k$ be a field, and let $X = \operatorname{Spec} k[x, y, z, w]/(xy−zw)\subseteq \mathbb{A}^4_ k.$ (a) Show that $D= V (x, z)$ is a prime (Weil) divisor on $X$ and that $\operatorname{Cl} X\simeq \mathbb{Z}$ is generated by the divisor class...
When is passing from real algebraic geometry to the complexification genuinely unavoidable?
Many results in real algebraic geometry are proved by passing from a real variety to its complexification , then studying the action of complex conjugation on . For example, one often regards $X(\mathbb R)$ as the fixed-point locus of conjugation on $X(\mathbb C)$. This appears in results such...
Non-openness of flat locus
If $f \colon X \to Y$ is a finite surjective morphism between integral Noetherian schemes, then the set $V\subseteq Y$ of points over which $f$ is flat is open. I want to show that this fails if we drop the finiteness assumption. I can think of an example given by blowing up $\mathbb A^3$ at a...
Title. Analytic validity of mapping the regularized sum of natural numbers to the sum of primes via a scalar product projection?
Title Analytic validity of mapping the regularized sum of natural numbers to the sum of primes via a scalar product projection Body In a recent preprint by Ezadiin Redwaan titled "A Scalar Product Approach to Strong Goldbach, Twin Primes, Polignac Conjectures And Geometric Unification of...
Question about Theorem 4.2 from Rational Points on Elliptic Curves
I have a question about a certain part of the proof of Theorem 4.2 from Rational Points on Elliptic Curves. Let $p$ be a prime so that $p=1\;\mathrm{mod}\;3$. Let $R=\{x^{3}|x\in\mathbb{F}_{p},x\neq 0\}$. Notation: $[XYZ]$ is the number of triples $(x,y,z)$ so that $x+y+z=0,x\in X,y\in Y,z\in...
Is there a completely elementary way to prove that $Y^2=X^3-32X$ has rank 1?
I’m working on a paper in which I end up considering the biquadratic rational curve $$u^2v^2 - u^2 - v^2 - 6uv + 8 = 0. \tag{$1$}$$ To complete the remainder of my proof/method, I need to prove that it has rank 1. I believe it can be transformed to the Weierstrass form $$Y^2=X^3-32X,$$ and then...
Near to Euler’s 4th power taxicab equation solution using $W^{4}+X^{4}=Y^{2}+Z^{4}$?
The above given equation solution is very easy just make it to an elliptic curve. For $$ W^{4}+X^{4}=Y^{2}+Z^{4} $$ Divide both sides $Z^{4}$, we wil get $$ \left(\frac {W}{Z}\right)^4 + \left(\frac{X}{Z}\right)^4 = \left(\frac{Y}{Z^2}\right)^2 +1 $$ If we substitute $\frac{W}{Z} = (u+v)$,...
Cubic Diophantine equation
Problem:I am looking for help with the following Diophantine equation: $$y^2 = x^3 - x^2 + 16$$ By working through the equation, I have successfully found 8 distinct non- negative integer solutions. The largest value of $x$ among all the solutions I found is $x = 112$ (which gives $y = 1180$)....
cubic unit with positive norm must be positive
Let $a$ be a positive integer, not a cube, so that $\alpha=\sqrt[3]a$ is irrational, and write $$R={\mathbb Z}[\alpha]=\{\,x+y\alpha+z\alpha^2\ |\ x,y,z\in\mathbb{Z}\,\}\ .$$ Let $\beta=x+y\alpha+z\alpha^2$ be a unit in $R$ with norm (product of conjugates) equal to $1$ (and not $-1$). Then...
Does Tate's $p$-adic uniformisation theorem hold over general non-archimedean local fields?
Tate's $p$-adic uniformisation theorem for elliptic curves goes as follows: Let $K$ be a $p$-adic field, let $E/K$ be an elliptic curve with $v_K(j) \ge 0$, and let $\gamma(E/K)=-c_4/c_6 \in K^{\times}/(K^{\times})^2$. a) There is a unique $q \in K^{\times}$ with $|q|<1$ such that $E$ is...