While studying parametric identities involving sums of three cubes, I obtained the polynomial identity in 11 variables. The complete formulas and verification code are available here: https://zenodo.org/records/22255821 All these identities have been checked symbolically in SymPy. The parametric...

Let $A\subset\mathbb{N}$ have positive upper density. Must there exist infinitely many $n$ such that both $n$ and $n^2$ belong to $A$? If not, determine the largest possible upper density of a set $A$ for which $$ A\cap\{n^2:n\in A\} $$ is finite. Can anyone please help me with this?

Throughout, $\mathfrak S_n$ is assumed to be associated with a rank-$(n-2)$ zeta function. Consider $\mathfrak{S}_3 = (\mathcal{I}_3, \Gamma_3)$, composed of four orbifold sheets glued along a seam graph $$ \mathcal{I}_3 = \left(\bigsqcup_{i=1}^4 X_i\right)\Big/\!\sim_{\Gamma_3}, \qquad X_i =...

Can someone relate the constants in the density function, of numbers whose digit sum divides its digit product to properties of these numbers? In attempting to solve Bernardo Recaman Santos' question on the Puzzling Stack Exchange, about triplets and quadruplets (in a row) of these numbers, as...

For the next multigrade diophantine chain $$x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k=z_1^k+z_2^k+z_3^k=N, k<5$$ This question is partly inspired by this similar question, but here we will look at degrees that are less than 5, which makes the discovery of new multi-grade solutions more likely....

Let $(a,b,c,d)$ be positive. In a previous question, we asked for parameterizations to, $$a^3+b^3+c^3 = (c+1)^3=d^3$$ where $c$ is a polynomial of deg-$n$ for $n>3$. Question: We've found $c$ of various deg-$n$, but now we give it sharper focus: Is there of deg-4 or deg-8? I. Degree 3 There are...

I have been using JSTOR to identify the original year, volume, issue, and problem number of problems that were cited only as “AMM” (American Mathematical Monthly) in various books. My approach has been to search for distinctive phrases and keywords from the problem statements and try to...

I'm interested in a particular relaxation of the magic square of squares problem, where the entries are of the form $a^2\pm\varepsilon$ for some $\varepsilon$. The original problem is when $\varepsilon=0$. By "correcting" the Parker square, there is a solution when $\varepsilon = 528$:...

Conjecture: Does the Diophantine equation $$x \cdot a^y + y \cdot b^z = z \cdot c^w + w \cdot d^x$$ And satisfying $$a + b + c + d = x + y + z + w$$ Also $$a \neq b \neq c \neq d \neq x \neq y \neq z \neq w$$ have any solutions in primes? Motivation: In a previous iteration of this problem,...

Motivation: While recently researching equations similar to this, I suddenly had an idea: would adding coefficients and specific restrictions change the difficulty? Therefore, I unexpectedly pieced together the following Diophantine equation: Let $a, b, c, d, x, y, z, w$ be 8 pairwise distinct...

The question is in the title: if you pick a natural number $j$, are you always guaranteed to have the fractional part of some $j^{k+1} \pi^k/2$ be strictly greater than $1/2$? One would certainly expect this to be true from normality/equidistribution-style properties, though those are...

Let $\pi_2(n)$ be the prime twin counting function. Let $li_2(x) = \int \frac{1}{\ln(t)}dx = li(x) - \frac{x}{\ln(x)}$ Let $C_2$ be the prime twin constant around $1.32$. Then we get the sharp conjecture for a prime $p > 5$ : $$|\pi_2(p) - C_2 li_2(p)| < \frac{\sqrt{p+2}}{3}+2$$ GENERALIZED...

This question was asked to me by a junior trying number theory problems and I got struck on it. Question:Show that there are infinitely many triples $(x,y,z)$ of integers such that $x^3+y^4=z^{31}$. I found a solution here https://www.isical.ac.in/~rmo/papers/rmo/rmo-2015-3.pdf : as problem $3$...

Background: I was exploring whether the standard normalization $\sum p_i = 1$ in probability is exact when the parts are written as real numbers. This led me to the quantity: $$\epsilon_n = \frac{1}{n}\left(1 - n^{-1/n}\right)$$ which represents a "gap" when dividing $1$ into $n$ equal parts....

Consider the reduced residue system modulo a primorial $$P_k=p_1p_2\cdots p_k.$$ Let $$R_k=\{r\in\{1,\ldots,P_k\}:\gcd(r,P_k)=1\},$$ ordered cyclically, and let $G_k$ be the cyclic sequence of gaps between consecutive elements of $R_k$. For example, for $$P_k=30=2\cdot3\cdot5,$$ the reduced...

I am a class 10 student exploring number theory and thinking about variations of famous problems like Legendre's conjecture.Let an interval between consecutive squares be defined for a given integer $n \ge 1$. Let $a_n$ represent the exact count of prime numbers that fall within the interval...

For any positive integer $n$, define $S(n)$ as the sum of the digits of $n$ in decimal (e.g., $S(19) = 1 + 9 = 10$). Define the sequence $a_k$ as: $a_1 = 7$ $a_{k+1} = a_k + S(a_k)$ Edit: To add, the first term ($a_1$) can be any positive integer(except $a_1 \not\equiv 0 \pmod 3$). I conjecture:...

This might be something trivial I missed but I was wondering. intro Let $w(n)$ be the number of distinct prime factors of the positive integer $n$. We know that $w(n)$ grows slowly. It seems logical to me that $$ \max(i<n,w(i))=A$$ $$\to w(n) < A+2$$ The simpler analogue conjecture for the count...

Let $p$ be a prime. Experimental data shows that for any $1 \le m < p$, $$ \lim_{p \to \infty} \left(\sum_{\substack{1\le a,b<p\\ab\equiv m\pmod p}} \frac{1}{b}\left(\frac{b}{a}\right)^{1/p} - \log p \right) = \gamma. $$ The result is not true if $p$ is composite. Can this be proved for...

Let the sequence $(a_n)$ be defined by $$ a_1=3,\qquad a_{n+1}=a_n^2-2 $$ for every positive integer $n$. Let $p_n$ be the smallest prime divisor of $a_n$. Prove that $$ p_n\ge 2n+3 $$ for every $n\ge 2$. Using mathematical induction, we prove that $ a_n=\alpha^{2^{n-1}}+\beta^{2^{n-1}} $ for...