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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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=...

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...

According to the book Number Theory II: Iwasawa Theory and Automorphic Forms by Nobushige Kurokawa, Masato Kurihara and Takeshi Saito, one way to prove the functional equation of the Dedekind eta function $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$ is as follows: Consider this :...

Following is an attempt at a proof (as an exercise) that the average of primes on an interval $[1,n]$ is asymptotically equal to the midpoint of the interval. While I think the idea is correct (see data below), there may be a better way, and notational improvements welcome. To show:...

Let $a, b$ be positive integers. In the Cartesian coordinate plane, consider the rectangular region $S$ (including the boundary) enclosed by the points $(1,1)$, $(1,b)$, $(a,1)$, and $(a,b)$. The line $$ l: (2a+1)y - (2b+1)x = 0 $$ divides $S$. Let $f(a,b)$ be the absolute value of the...

Let $\lambda(n) = (-1)^{\Omega(n)}$ be the Liouville function and $$ C(x) = \sum_{n \le x} \frac{\lambda(n)}{n}. $$ By Turán's classical approach, $C(x)$ is controlled by $$ \sum_{n=1}^\infty \frac{\lambda(n)}{n^{s+1}} = \frac{\zeta(2s+2)}{\zeta(s+1)}, $$ and each nontrivial zero $\rho = \beta +...

This question was asked in my assignment and I am stuck on it. Question:If $A$ is infinite integral domain , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then show that $G=0$. Attempt: Let $D(F)= A^I \setminus V_A(F)=${$a\in A^I : F(a)\neq0$} . Given...

I realize this may be an unusual request, but I am trying to find out whether this style of studying mathematics already exists, or whether there are resources that come close to it. I am not looking for a standard textbook on Galois theory, nor for a historical overview followed by the modern...