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Conjecture: Let $\mu_x$ be the arithmetic mean of the ratio of the perimeter to the hypotenuse of all primitive Pythagorean triplets in which no side exceeds $x$; then, $$ \lim_{x \to \i …

I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO. While answering another question in MSE I …

I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime. $$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k $$ At $a=1$, this …

This question was posted in MSE. It got many upvotes but no answer hence posting it in MO. A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a valu …

The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le …

Is it true that if $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then it has a prime factor greater than $17$? Note: This question was asked in MSE but did not receive an answ …

If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call these …

Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that $$ \sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0 $$ My experimental data for $n \le 6 \times 10^5 $se …

Note: Posting in MO since it was unanswered in MSE Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \l …

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1. I …

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has …

What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients $$ S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p} $$ where the sum runs o …

We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a tria …

In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series expan …

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it …