共 345 个问题，第 6/18 页
embedding of $SL_2$ into larger matrix groups that decreases coefficient size?
Are there injections $\phi:SL_2(\mathbb{Z})\to M^{n\times n}(\mathbb{Z})$ that decrease the size of the coefficients? For the application I have in mind, coefficient growth is exponential in the word length w.r.t. a generating set, e.g. the maximum entry of words of length $n$ in the generators...
A question about the proof of Tate's algorithm in ATAEC
(I'm sorry for my English.) Hello. I have been reading the book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman. In the course of the proof of Tate's algorithm (page 375, at the end of the proof of Step 8), there is an equality on the order as follows:...
$\mathbb{C}$ is closure of residue field modulo infintely large prime
It is a well-known fact that there exists a non‑principal ultrafilter $\mathcal{U}$ on the set of prime numbers such that the ultraproduct of the algebraic closures of the finite fields is isomorphic to the complex numbers: $$ \mathbb{C}\;\cong\; \prod_{\mathcal{U}}\ \overline{\mathbb{F}}_p . $$...
Gal sum for square free integer
I have a question regarding this paper by Tenenbaum and Bréteché. They define $$ S_\alpha(\mathcal{M}) = \sum_{m,n\in\mathcal{M}} \frac{(m,n)^\alpha}{[m,n]^\alpha} = \sum_{m,n\in\mathcal{M}} \biggl( \frac{(m,n)^2}{mn} \biggr)^\alpha \quad\text{and}\quad \Gamma_{\alpha}(N) =...
Binary quadratic forms in $\Bbb Z_2$
$\def\Z{{\Bbb Z}}\let\f\frac\let\d\delta$Concerning binary forms, there is something I don’t understand: Let $F$ be a form $F=aX^2+2bXY+cY^2$, $a,b,c\in\Z$ or $\Z_2$. Let assume $F$ to be primitive, that is $a$ or $c$ odd, and so invertible (or unit) in $\Z_2$. I write $$F=a\left(X+\f...
Exponents of Mersenne primes represented by the quadratic form $x^2+dy^2$
List of Mersenne prime exponents: [2,3,5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,...
Should we expect an approximate version of Euler’s Sum of Powers Conjecture to hold for higher power equations?
Consider the Diophantine equation: $$x_1^k+x_2^k+\dots+x_n^k=y^k.\tag 1$$ For any exponent $k$, let $N(k)$ be the smallest possible number of terms on the left, $n$, such that there exists a non-trivial solution in positive integers. Euler’s Sum of Powers Conjecture is equivalent to the...
An Elliptic curve for solving $W^4+X^4=Y^4+Z^4$
It is well known that a parameterization for the equation $$W^2+X^2=Y^2+Z^2$$ is given by: $$(W,X,Y,Z)=(a+b,ab-1,ab+1,a-b).$$ I have found a similar type of parameterization for the fourth-power equation: $$W^4+X^4=Y^4+Z^4$$ where $W = d+e$, $X = cde-1$, $Y = d-e$, and $Z = cde+1$, provided that...
Is there a special name for morphisms sharing some special condition?
Let $\Phi=[F_1,\ldots,F_N]$ be a map, with $F_i$ homogenous polynomials of variables $X_1,\ldots,X_N$ of the same degree with integer coefficients. The map $\Phi$ mapping $Z^N$ into itself may share the following property: Let $P=(x_1,\ldots,x_N)\in Z^N$ be any point such that $\gcd...
Can you express n-th degree roots as n-th roots and n-sections?
I have zero, one, addition, subtraction, multiplication, division (by a non-zero expressible number), n-th roots (of positive expressible numbers, where n is expressible), and all the trigonometric functions (in their default domain restricted to the expressible numbers). The number expressible...
How does undergraduate or real math research papers differ from research papers written by high-schoolers at ISEF
I'm a high school freshman with a strong passion for mathematics. I'm currently studying number theory through a math circle, and I'm particularly interested in pursuing mathematical research in areas such as Egyptian fractions, prime numbers, Catalan numbers, and related topics. I've been...
Can 1 be written as a finite sum of distinct unit fractions from any arithmetic progression?
Let $a,d$ be positive integers. Is there a reasonably short elementary proof that one can find distinct nonnegative integers $n_1,\dots,n_k$ such that $$ \frac1{a+n_1d}+\frac1{a+n_2d}+\cdots+\frac1{a+n_kd}=1? $$ Equivalently, can one choose finitely many distinct terms of every infinite...
How to prove that following set is closed
I am self studying Algebraic geometry from Gortz and Wedhorn's Algebraic Geometry :1 Schemes. I have a question on Page $16$ of the textbook just after the definition of morphism of affine algebraic sets. Remark $1.29$: the definition (of morphisms between affine algebraic sets) shows that a...
Question in Proposition $1.40$ of Algebraic Geometry $1$ by Gortz and Wedhorn
I am unable to understand the proof of proposition $1.40$ given on Page $21$ of the textbook by Gortz and Wedhorn. Definition $1.30$ Let $X\subset \mathbb{A}^n{k}$ be the affine algebraic set The $k-$algebra $\Gamma(X)= k[T_1,...,T_n]\cong Hom (X, \mathbb{A}^1(k))$ is called the affine...
Inconsistency in Galois Theory?
I am a high school student who, after some tinkering, came across the Abel–Ruffini theorem. I then learned about its explanation through Galois theory, particularly the result that a polynomial is solvable by radicals if and only if its Galois group is solvable. This leads to a confusion. The...
Can we say something about primes $p$ s.t. $p^p-2$ is prime?
So far I have found that 2 and 7 are such primes. However, due to the exponential form, I can't compute very far for more examples (currently, get stuck at 19). More specifically, I want to know whether there exists a finite or infinite amount of these primes. I am not well versed in number...
Combinatorial interpretation of the integer $\frac1{n!}b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)$, for integers $a$, $b$, $n$ (with $n>0$)
On the IMO 1985 Longlist problem 11, it is asked to prove that $$\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}$$ is an integer, where $a$, $b$, $n$ are integers, and $n>0$. The expression resembles a binomial coefficient and seems to have some combinatorial meaning. What would that be?
How to prove that $\operatorname{Hom}_{Var} (X,Y) \cong \operatorname{Reg}(X,Y)$
I am self studying algebraic geometry from the textbook of Daniel Perrin (Algebraic Geometry: An Introduction). On page 44 is the Proposition 3.5 which I am unable to prove and need help with. Proposition 3.5. Let $(X,O_X)$ and $(Y,O_Y)$ be two affine algebraic sets equipped with the affine...
Show that a non -empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other.
I have a question in the proof ofCorollory $4.4$ of Daniel Perrin's Algebraic Geometry on Page $45$. Corollary $4.4$ A non empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other. Proof:By quasi-compactness, we can write $X=...
$2$ questions in proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin(page $46$)
I have 2 question in the proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin.( Page $46$). Statement of Proposition $4.6$: Let $X$ be an algebraic variety and let $Y$ be a closed set in $X$. We define a sheaf of rings $O_Y$ of $Y$ by setting $O_Y(V)= ${$f:V\to k| \forall x\in V...