Articles of conjectures

If $q^k n^2$ is an odd perfect number with Euler prime $q$, which of the following relationships between $q^2$ and $n$ hold?

(Preamble #1: In what follows, we take $\sigma=\sigma_{1}$ to be the sum of the divisors, and denote the abundancy index of $x \in \mathbb{N}$ as $I(x)=\sigma(x)/x$.) (Preamble #2: My sincerest apologies for the somewhat very long post — I just had to put in all the details into one place for ease of quick reference […]

If $N=q^k n^2$ is an odd perfect number and $q = k$, why does this bound not imply $q > 5$?

Let $\mathbb{N}$ denote the set of natural numbers (i.e., positive integers). A number $N \in \mathbb{N}$ is said to be perfect if $\sigma(N)=2N$, where $\sigma=\sigma_{1}$ is the classical sum of divisors. For example, $\sigma(6)=1+2+3+6=2\cdot{6}$, so that $6$ is perfect. (Note that $6$ is even.) Denote the abundancy index of $x \in \mathbb{N}$ as $I(x)=\sigma(x)/x$. Euler […]

What is wrong with this proof that there are no odd perfect numbers?

Main Question What is wrong with this proof that there are no odd perfect numbers? The “Proof” Euler proved that an odd perfect number $N$, if any exists, must take the form $N = q^k n^2$ where $q$ is the Euler prime satisfying $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$. Denote the sum […]

Does this category have a name? (Relations as objects and relation between relations as morphisms)

Given two relations $R\subseteq A\times B$ and $R’\subseteq A’\times B’$. Is it known/used that every relation $r\subseteq R\times R’$ can be characterized by two relations $\alpha\subseteq A\times A’$ and $\beta\subseteq B\times B’$ so that $((a,b),(a’,b’))\in r \iff \Big((a,a’)\in\alpha\wedge (b,b’)\in\beta\wedge (a,b)\in R\implies (a’,b’)\in R’\Big)$ and if $R”\subseteq A”\times B”$, $\,r’\subseteq R’\times R”$, where $r’$ is characterized […]

Prove that $\lim\limits_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0$.

I conjecture that for any $\epsilon>0$, we have $$ \lim_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0 $$ where $\Gamma(x,a) = \int_a^\infty t^{x-1}e^{-t} \mathrm{d}t$ denotes the incomplete Gamma function, and $\Gamma(x) = \Gamma(x,0)$ is the Gamma function. I’m also happy with the weaker conjecture $$ \lim_{x\to\infty} \frac{\int_{x(1+\epsilon)}^\infty t^{x-1}\log(1+t)e^{-t} \mathrm{d}t}{x\Gamma(x)}=0 $$ but I doubt that it is any easier. Any clues? NB: […]

How often does $D(n^2) = m^2$ happen, where $D(x)$ is the deficiency of $x$?

Let $\sigma=\sigma_{1}$ denote the classical sum of divisors. For instance, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$. Let $x \in \mathbb{N}$ ($\mathbb{N}$ is the set of natural numbers/positive integers). Denote the number $2x – \sigma(x)$ by $D(x)$. We call $D(x)$ the deficiency of $x$. Now, let $m, […]

The conjecture that no triangle has rational sides, medians and altitudes

I have found a conjecture that there is no triangle whose sides, medians, altitudes and area are all rational. I figure that someone must have already found such a triangle if one existed and yet I saw this conjecture but haven’t come across any proof or hints. Is this still an open question? I feel […]

Disprove the Twin Prime Conjecture for Exotic Primes

The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form $2^p – 1$ where $p$ is a prime, A000668, $43\%$) Sophie Germain primes ($p$ and also $2p+1$ is prime, A005384, $42\%$) Fermat primes (of the form $2^{2^k} + 1$, A019434, $100\%$) regular primes (A007703, $61\%$) Fibonacci […]

Measure Theory Conjecture

While I was doing some math here, I made this conjecture. Let $f_n:X\rightarrow \mathbb{R}$ be a sequence of measurable functions from the measure space $(X,\mathcal{A},\mu)$ to the measurable space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and let $\mu:\mathcal{A}\rightarrow[0,\infty]$ be a positive measure. Consider that for all $\varepsilon > 0$, the sets $A_n = \{x\in X: \ |f_n(x)| > \varepsilon\}$ are […]

making mathematical conjectures

If a non-mathematician wanted to conjecture something and had strong numerical evidence to support the conjecture, how would he/she go about doing so? Would the mathematical community (a) take it seriously? (b) even look at it?