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 +...

My name is Kadin Shah, and I am a rising senior at Arizona State University studying Applied Mathematics with an emphasis in mechanical engineering. I'm interested in using creative problem-solving solutions in a collaborative setting with engineers at a large company such as a National...

I had currently passed 10th grade. I want to go in pure mathematics and research and publish paper in that field (especially Number Theory) please tell me roadmap and the books to follow one after one. Please Guide me.

For even number $n$, if $n$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is not in OEIS sequence A252502, but if $n$ is a totient, must be $n$ in A252502? For odd number $n$, if $n-1$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is...

Erdős Problem 252:(https://www.erdosproblems.com/252) Let $k\geqslant 1$ and $\sigma_{k}(n):= \sum_{d|n} d^{k}$. Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!}$ irrational? Currently this problem is open for $k \geqslant 5$. I am considering a different version of this problem...

In this related question, @Gerry Myerson asked if the conjecture fails for composites. Upon examining for composites, I found a pattern for module $n^2$ instead of $n^3$ in the linked original question. I have verified these conjectures for $n \le 5 \times 10^6$. Can they be proved or disproved....

My experimental data suggests that the following binomial sum congruence involving primes and Fibonacci numbers hold. I have experimentally verified them for all primes $\le 1.5 \times 10^6$. Can they be proved or disproved? Conjecture 1. Let $p$ be a prime, $ p\equiv 1 \text{ or } 19 \pmod{30}....

I. Question Consider the following independent and symmetric mixed equations, $\quad x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4$ $\quad x_1^8+x_2^8+x_3^4+x_4^4 = y_1^8+y_2^8+y_3^4+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^4 = y_1^8+y_2^8+y_3^8+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^8+x_5^4 =...

I am investigating the Diophantine equation $$x^4 - Ny^4 = 1$$ where $N$ can be expressed as a sum of three fourth powers: $$N = a^4 + b^4 + c^4$$ for integers $a, b, c$. Background: I see some from $\mathrm{A356496}$ for squarefree integers $k$ such that $x^4 - ky^2 = 1$ has a nontrivial...

Title: Primes as $X-Y$ with $\gcd(X,Y)=1$ and $\operatorname{rad}(XY)$ equal to the product of all primes below $p$ — is this known? While experimenting with multiplicative decompositions of primes, I arrived at the following question. I would like to know whether it is already in the...

I found the following infinite continued fraction: $$ \frac{1}{G} = \frac1{(2\pi)^{-3/2}\,\Gamma^2(1/4)} = \cfrac{4}{1+\cfrac{5}{1+\cfrac{6}{1+\cfrac{9}{1+\cfrac{8}{1+\cfrac{13}{\ddots}}}}}} $$ where $G$ denotes Gauss's constant, and the partial numerators are defined by interweaving sequences...

I've been experimenting with a self-similar construction for the Collatz map (the $n \to n/2$ / $n \to 3n+1$ function) and found a pattern I can partially — but not fully — explain. I'd appreciate a sanity check and any pointers to relevant literature. Construction. Fix an odd integer $x$ with...

While creating quantitative aptitude problems for management entrance exam preparation, I noticed that the Euclidean algorithm is almost always preferred over prime factorization for computing the greatest common divisor.

I am in the middle of a problem which needs showing that the following system of congruences has finite number of solutions. I have verified up to some extent through sage that this has only two solutions for $(q, r)$ (with $q<r$) namely $(11, 17)$ and $(23, 103)$. The system of congruences is...

Is it possible to know how many steps are left to reach 1 knowing only x? The main idea is: when we analyze numbers of the form $n = 3^x + 2^x$ (for $x \ge 1$), we can track the Collatz trajectory using algebra instead of doing it number by number. Following the rules (if it is odd, multiply by...

Consider the digit dynamical system $$ f(n)=\bigl(E(n)\,O(n)\bigr)^2, $$ where $E(n)$ and $O(n)$ denote the sums of the even and odd decimal digits of a positive integer $n$, respectively. The state of an integer may be represented by the ordered pair $$ (E(n),O(n)). $$ I plotted all attainable...

I have been analyzing the problem of integer factorization by looking at non-trivial square roots of unity modulo $n$. Starting directly from the quadratic congruence $x^2 \equiv 1^2 \pmod n$, I derived a specific parameterization that maps the problem onto a generalized Pell equation of the...