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 would like to propose a prime-based variant of the Collatz conjecture that I came up with. I am interested to know if this specific variation has been studied before, or if there are any known counterexamples to the behavior I observed. Definition of the Function Let $n > 1$ be a positive...

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,...

I've been having fun trying to find new number systems that aren't in the OEIS. One such number system, is the "Snowball Numbers", which I will define below. Apologies if these have been explored before, I could not find them. While playing with these numbers, I found the Catalan numbers (?!)...

Let $p_k$ denote the $k$-th prime number, and let $\operatorname{rev}(x)$ denote the decimal digit-reversal of a positive integer $x$. Define the consecutive-prime geometric sums: $$A_n = \sum_{i=0}^{p_n} p_{n+1}^i = \frac{p_{n+1}^{p_n + 1} - 1}{p_{n+1} - 1}, \qquad B_n = \sum_{i=0}^{p_n}...

I have been investigating an empirical variant of the classical Josephus problem and would like to know whether it has been studied previously. Let $J(n,k)$ denote the survivor of the classical Josephus problem with population size $n$ and elimination interval $k$. For fixed $n$, instead of...

Background & Definition In lattice geometry, two points $A, B \in \mathbb{Z}^2$ are said to be visible to one another if the open line segment between them contains no other lattice points. Equivalently, if $A = (x_A, y_A)$ and $B = (x_B, y_B)$, they are visible if $\gcd(|x_A - x_B|, |y_A -...

I have been analyzing a recursive sequence based on the greatest common divisor that acts as a dynamic sieve for twin primes. It shares structural similarities with Rowland's prime-generating sequence but targets the difference of squares. For any integer $n \ge 2$, define the sequence...

The Fermat quintic threefold is given by the equation, $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5=0\qquad\qquad$$ $\hskip1.5in$ (Incidentally, this threefold is a Calabi-Yau manifold, a type of manifold important to string theory.) In the integers, there are only four primitive solutions known, two which...

I am exploring a family of integer sequences $(k_n)$ that combine explosive quadratic growth with specific "modular" reduction rules based on the proximity to perfect squares and powers of two. I’ve categorized these reduction mechanisms as "Valves." The Core Growth Function: Let $f(x) = ax^2 +...

Let $(a,b,c)$ be positive integers such that a $abc+1 \mid a^2+b^2+c^2$. Then $\dfrac{a^2+b^2+c^2}{abc+1}$ can be written as the sum of 2 positive squares. Proposed by Sam Vandervelde.

Professor Manin's 1990 ICM talk states there is a convincing case to be made that these groups are, in some sense, "more fundamental" in number theory than even the integers. The talk's purpose being a broadest-possible survey of the works of Professor Drinfel'd, Professor Manin does not deem it...

Let $$ t_n = {16 \over (2n+1)5^{2n+1}} - {4 \over (2n+1)239^{2n+1}}. $$ Define $$ S_N = \sum_{n=0}^N (-1)^n t_n. $$ For each integer $m \geq 0$, define $$ I_m = [S_{2m+1}, S_{2m}]. $$ By Machin's formula, $$ \pi = 16\arctan(1/5) - 4\arctan(1/239), $$ so $$ S_{2m+1} \leq \pi \leq S_{2m}. $$ Also,...

This is extended computational evidence for the Conjecture from this question, focusing on the $n=1$ case. Conjecture A for $n=1$ states: for all sufficiently large primes $p$ with $q \mid p-1$, $$v_p\!\left(\prod_{k=1}^{p} \Phi_q(k)\right) = q-1.$$ Reduction to $v_p(k^{p-1}-1)$. By the...

Let $N_n=\binom{2n}{n}^3(42n+5)$. Define $L_m=\sum_{n=0}^m \frac{N_n}{2^{12n+4}}$ and $U_m=L_m+\frac{4N_{m+1}}{3\cdot 2^{12m+16}}$. These are rational intervals coming from a Ramanujan series for $1/\pi$. For this question, I only want to study rational points inside these explicit intervals....

In Kedlayas lecture notes (https://kskedlaya.org/prismatic/sec_overview.html) about prismatic cohomology he introduces the ring of Witt vectors using $\delta$-rings via the fact that the Witt vector functor $W$ is the right adjoint to the forgetful functor $\mathbf{Ring}_{\delta}\to...

For a semiprime $N = p×q$, consider the sequence $f(k) = k^2 + k + N$, for $k = 0, 1, 2, ...$ Since $f(k) ≡ k(k+1) \mod N$, we have gcd($f(k), N$) = gcd($k(k+1), N$). This means a factor of $N$ is revealed at position $k$ whenever $p$ divides $k$ or $k+1$. For $N = 77 = 7×11, f(k) = k^2 + k +...

While trying to prove FLT for $n=7$, I came to find that there could be a certain class of solutions (counterfactual) for the counterfactual case that FLT is false for $n=7$: $(d^7+x^7+y^7)^7 = (d^7+x^7-y^7)^7 + (d^7-x^7+y^7)^7$ where $x, y$ and $d$ are coprime and all of them are coprime to...