My experimental observation suggests that the sum of the reciprocals of the square roots of the products of all multiplicative-inverse pairs modulo a prime $p$ tends to $5$ as $p \to \infty$. More specifically, for each prime $p$, consider the pairs $(a,b)$ satisfying $1\le a,b\le p-1$ and...

Let $T_n = \frac{n(n+1)}{2}$ denote the $n$-th triangular number, and let $S(m)$ denote the sum of the digits of $m$ in base 10. I am investigating a strong version of a digit-sum conjecture: Does $S(T_n)$ yield infinitely many distinct prime numbers? Modular and Growth Behavior We know that...

The problem was motivated by this related MSE question, although the recurrence here is structurally different. Let $f, g : \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ satisfy $$f(n) \ge n+1, \qquad g(n) \ge (1+\varepsilon)n$$ for some fixed $\varepsilon > 0$. Consider $$a_{n+2} = a_{n+1} \bigl(1 +...

We are to prove that if $n$ is fixed natural number then exists arithmetic $a_n$ and geometric $b_n$ sequences of integers that: $$b_1 < a_1 < b_2 < a_2 < \ldots < b_n < a_n.$$ Sketch It’s equivalent to construct such sequences of fractions – we can always multiply by such large $N$ as is...

I observed experimentally that for every positive integer $a$, the interval $$ (a,11a/5] $$ seems to contain at least $\lfloor\sqrt a\rfloor$ primes. Equivalently, if $\pi(x)$ denotes the prime-counting function, the claim is $$ \pi(11a/5)-\pi(a)\ge \lfloor\sqrt a\rfloor $$ for every positive...

Semiprime Covering of Residue Classes Modulo a Primorial Let $p$ and $q$ be consecutive primes, with $q$ the smallest prime greater than $p$, and let $p\#=\prod_{\ell\le p,\ \ell\text{ prime}}\ell$ denote the primorial of $p$. Let $\mathcal S$ denote the set of semiprimes, $\mathcal...

This is in regards to building the DAG and showing the structural organization of the system rather than the superficial view of the Syracuse map

Let $p$ be an odd prime and $a\geq 1$. Consider the reduced residue system modulo $p^a$, namely the set of integers $x$ such that $$ 1\leq x\leq p^a,\qquad \gcd(x,p)=1 $$ I would like to determine the following power sum modulo $p^a$: $$ S_k=\sum_{\substack{1\leq x\leq p^a\\ \gcd(x,p)=1}}x^k...

Are there injections $\phi:SL_2(\mathbb{Z})\to M^{n\times n}(\mathbb{Z})$ that decrease the size of the coefficients? For the application I have in mind, coefficient growth is exponential in the word length w.r.t. a generating set, e.g. the maximum entry of words of length $n$ in the generators...

$\def\Z{{\Bbb Z}}\let\f\frac\let\d\delta$Concerning binary forms, there is something I don’t understand: Let $F$ be a form $F=aX^2+2bXY+cY^2$, $a,b,c\in\Z$ or $\Z_2$. Let assume $F$ to be primitive, that is $a$ or $c$ odd, and so invertible (or unit) in $\Z_2$. I write $$F=a\left(X+\f...

List of Mersenne prime exponents: [2,3,5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,...

Consider the Diophantine equation: $$x_1^k+x_2^k+\dots+x_n^k=y^k.\tag 1$$ For any exponent $k$, let $N(k)$ be the smallest possible number of terms on the left, $n$, such that there exists a non-trivial solution in positive integers. Euler’s Sum of Powers Conjecture is equivalent to the...

I'm a high school freshman with a strong passion for mathematics. I'm currently studying number theory through a math circle, and I'm particularly interested in pursuing mathematical research in areas such as Egyptian fractions, prime numbers, Catalan numbers, and related topics. I've been...

Let $a,d$ be positive integers. Is there a reasonably short elementary proof that one can find distinct nonnegative integers $n_1,\dots,n_k$ such that $$ \frac1{a+n_1d}+\frac1{a+n_2d}+\cdots+\frac1{a+n_kd}=1? $$ Equivalently, can one choose finitely many distinct terms of every infinite...

So far I have found that 2 and 7 are such primes. However, due to the exponential form, I can't compute very far for more examples (currently, get stuck at 19). More specifically, I want to know whether there exists a finite or infinite amount of these primes. I am not well versed in number...

On the IMO 1985 Longlist problem 11, it is asked to prove that $$\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}$$ is an integer, where $a$, $b$, $n$ are integers, and $n>0$. The expression resembles a binomial coefficient and seems to have some combinatorial meaning. What would that be?

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...