Let $$ \gamma=\lim_{x\to\infty}\left(\sum_{n=1}^x \frac1n-\log x\right) = 0.57721... $$ be Euler’s constant, and let $$ M=\lim_{x\to\infty}\left(\sum_{p \text{ prime}}^{p\le x}\frac1p-\log\log x\right)=0.26149... $$ be Mertens’ constant. These are two examples of reciprocal sums with (iterated-)...

Let all terms be positive. There are infinitely many solutions to, $$a^2+b^2 = (b+1)^2$$ $$a^3+b^3+c^3 = (c+1)^3$$ like the well-known $3^2+4^2 = 5^2$ and $3^3+4^3+5^3 = 6^3$. However, this has versions for higher degrees. For $4$th powers by Jaroslaw Wroblewski, $$178^4 + 1345^4 + 10400^4 +...

In the recent IMO 2026 Problem 1, a blackboard game is played where two active integers $m, n > 1$ are repeatedly selected and replaced by: $$g = \gcd(m, n) \quad \text{and} \quad l' = \frac{\operatorname{lcm}(m, n)}{\gcd(m, n)}$$ It is a known result that this game must terminate in a finite...

I have been studying the following digit dynamical system. Let $E(n)$ denote the sum of the even digits of $n$, and let $O(n)$ denote the sum of the odd digits of $n$. Define $$ f(n)=\big(E(n)\,O(n)\big)^2. $$ I am interested in the dynamical system obtained by repeated iteration $$...

Let $G=\operatorname{Spec}(R)$ be a finite flat commutative group scheme over $S=\operatorname{Spec}(A)$ of characteristic $p>0$. Suppose the $p$-Lie algebra $\operatorname{Lie}(G/S)$ is locally free. I would like to know the $p$-mapping on $\operatorname{Lie}(G/S)$ coincides with differential...

for the diophantine $x_1^8+x_2^8+x_3^8+x_4^8=y_1^8+y_2^8+y_3^8+y_4^8 $ Back in 2006, Nuutti Kuosa discovered the following non-trivial integer solution $$1953^8+2012^8+3113^8+861^8=1128^8+2767^8+2557^8+2823^8$$ A system of equations was used to speed up the search....

I am working on a sentence of Yves Hellegouarch in his book "Invitation aux mathématiques de Fermat-Wiles" . In the Fermat section, page 38, he tells that Fermat probably associated the equation $z^p=x^p+y^p$ to the form $X^2+(-1)^{(p+1)/2}pY^2$. I don't really understand the reason he thinks...

Let $\mathcal P$ denote the set of prime numbers, and consider $$ N =\frac{1}{1-\sum_{p\in\mathcal P}2^{-p}}. $$ Numerically, the binary expansion of $N$ appears to behave like that of a base-$2$ normal number. For example, among the first $10^6$ binary digits, the frequencies of $0$ and $1$,...

In the rather new MSE post Recursive sequence $a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$ generating twin primes, a sequence of integers $(a_k)_{k \geq 1}$ is associated to each natural number $n$, namely $a_0 = n^2$ and $a_{k+1} = a_k - \gcd(a_k,(n+k)^2-1)$. What is interesting, as pointed out by...

I find it easiest to explain the motivation as a "game" or task: you are given an arbitrary (but guaranteded to be valid) subsequence of the characteristic function of primes, so just '1's and '0's, one after another, and your task is to identify the numbers they represent. To simplify, let's...

I am an independent researcher investigating the arithmetic and structural properties of prime distributions formulated within the twin-track lattice of $6n \pm 1$. I would like to inquire about a potential algebraic and geometric invariant regarding Goldbach pairs. Consider the following model...

Determine all monic polynomials $P(x)$ with integer coefficients for which there exists a monic polynomial $Q(x)$ with integer coefficients such that, for every pair of positive integers $m,n$, $P(m^2+mn+n^2)\ne 0$ and $$ P(m^2+mn+n^2)\mid Q(m^4+m^2n^2+n^4). $$ Let $$ A=m^2+mn+n^2,\qquad...

(I'm sorry for my English.) Hello. I have been reading B.Mazur's article "An introduction to the deformation theory of Galois representations". I'm at the proof of proposition1 in §30, where he discusses I-ordinary deformation. Let me write down the settings : $A$ : a Noetherian local ring which...

I. Manifolds A non-singular homogeneous polynomial of degree $n+2$ with $n+2$ variables is a compact Calabi-Yau manifold, some of which important in string theory. For $n=2,3,5$ dimensions, we have, $$x_1^4+x_2^4+x_3^4+x_4^4 = 0$$ $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5 = 0$$...

Let $a(b,n)$ be the number of integer tuples $(x_1, x_2, ..., x_{k+1})$ where $0 \leq x_i \leq b-1$, such that $|x_i - x_{i+1}| = d_i$ for all $i$, where $(d_1, d_2, ..., d_k)$ are digits of $n$ in base $b$. Related patterns in this specific sequence are discussed here and here. Now consider the...

It seems that for some integer $N$, namely any $N\equiv4\pmod {16}$, then they can be expressed as sum of $4$th powers of $4$ positive rational numbers. For example: $$15236 =\left(\frac{1875}{251}\right)^4+\left(\frac{11767}{3263}\right)^4+...

I have designed a new variant of the Collatz conjecture based on modulo 4 remainders, and I am looking for computational data or heuristic analysis regarding its convergence. Definition of the Mapping Let $n$ be a positive integer. We define the transition function $g(n)$ as follows based on $n...

Consider the recurrence $$ x_{n+1}=x_{n-1}+2\log x_n, $$ with positive initial values chosen so that the sequence remains positive and increasing. Since this may be rewritten as $$ \frac{x_{n+1}-x_{n-1}}{2}=\log x_n, $$ it resembles the centred-difference discretisation of the differential...

Suppose $a(n)$ is the minimum integer $k$ such that $k^{11}$ can be expressed as the sum of $n$ distinct positive 11th powers. Q: Find $a(n)$ for $10\le n\le 19$. For example, $a(20)=119$ because $199^{11}$ is the sum of 20 terms of $11$th powers,...