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...

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ be infinite arithmetic progressions of integers. We consider the set of all second-order quadratic recurrence relations: $$s_{n+1} = a s_n^2 + b s_n s_{n-1} + c s_{n-1}^2 + d s_n + e s_{n-1} + f$$ where $(a, b, c,...

I started from a modular experiment for integers of the form $$ N=2k+11, $$ where I iterated $$ U_0=k+1,\qquad U_{n+1}=U_n^2-k\pmod N, $$ and called an integer $N$ captured if the orbit reached a solution of $$ x^2\equiv k\pmod N. $$ After rewriting the iteration symbolically using $$...

Here is a quite interesting problem I've come up with: Let $\mathbb{N}^+ = \{1, 2, 3, \dots\}$. Does there exist a sequence of sets $A_1, A_2, \dots$ such that: $1$. $A_k \subset \mathbb{N}^+$ is nonempty and finite. $2$. $A_i \cap A_j = \varnothing$ for $i \neq j$. $3$. $\bigcup_{k=1}^{\infty}...

Its easy to show that there are infinite amount of two consecutive numbers $n, n+1$ such that in their prime factoring all primes are in power at least two. It is because if one have such $n, n+1$ then construct another $(2n+1)^2 - 1, (2n+1)^2$ ; we start with $(288,289)$. But are there three...

Like say 2 is 7, 4, 13, 7, 4, 13, ... Again for 3, for 4 and so ob. It always ends in this 7,4,13 loop. Now we need to prove whether it will always end in this loop or not.

I need a source for the following statement: Let $q(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with $a,b,c\in\mathbb Z$ and let $p$ be a prime with $p\not\mid 2D$, where $D=b^2-4ac$ is not a square. Then the quaternary quadratic form $q(x_1,y_1)-q(x_2,y_2)$ represents all $p$-adic integers,...

I found the following statement (in different words with different functions) on another website: $$ \sigma(n)=2\left(n+\sum_{\substack{d|n\\d<\sqrt{n}}}d\right) -1 $$ if and only if $n=392.$ That $392$ is a solution is easy to check. Whether there are other solutions depends on "the first half"...

I would like some help with this question: let $S$ be a sparse subset of $\mathbb {N }$. Let $M$ be a subset of $S$ such that if $m\in M$ and $sa=m$ with $s\in S$ implies that $s=m$ and $a=1$. Let $S(x)$ be the number of represnetations of elements of $S$ less than x. We say that $c(n)$ is the...

Recently we encountered a difficult problem in computer science, but since it is very closely related to mathematics, I was unsure which board would be more appropriate. In the end I posted it here on the mathematics board. To make the problem easier to understand, I will give both a...