(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 […]

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 […]

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 […]

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 […]

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: […]

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, […]

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 […]

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 […]

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 […]

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?

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