# Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

954 questions
Filter by
Sorted by
Tagged with
132 views

114 views

### Number theoretical function with logarithmic properties

Introduction This is all about the function $v_a(x)$, what i could find about it and what questions i still have. First of all, I define $v_a(x)$ as the function that counts how often $x$ is divisible ...
291 views

### What methods show that a number is transcendental?

I've been doing a lot of research on such theories lately and these are all I've found so far: Liouvilles criterion (here) Lindemann-Weierstrass theorem (here) Gelfond-Schneider theorem (here) ...
154 views

### On Composite Numbers of the Form $p_{1}p_{2} \ldots p_{k} - 1$

This question is related to D. H. Lehmer's 1932 conjecture on Euler's totient function: Are there any composite $n$ for which $\phi(n)$ divides $n-1$? See, for example: On Lehmer's Totient ...
174 views

### Does there exist some prime $k$ for which there will be exactly two primes of the form $n!+k$?

This is a question related to my recent question Conjecture: “For every prime $k$ there will be at least one prime of the form $n!\pm k$” true? Using PARI/GP I searched for the number of primes of ...
4k views

### What do mathematicians mean when they say some conjecture can’t be proven using the current technology?

When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What ...
538 views

### Conjecture: “For every prime $k$ there will be at least one prime of the form $n! \pm k$” true?

Using PARI/GP, I searched for primes of the form $n!\pm k$ where $k \ne 2$ is prime and $n\in \Bbb{N}$. With the help of user Peter, we covered a range of $k \le 10^7$ and couldn't find a prime $k$ ...
990 views

### Conjecture: Is $100$ the only square number of the form $a^b+b^a$?

Conjecture $100$ is the only square number of the form $a^b+b^a$ for integers $b>a>1$. In other words, $(a,b)=(2,6)$ is the only solution. Can we prove/disprove this? Observations The ...
69 views

### If a conjecture suggests $P(x)$ is true for all $x$ in an infinite domain, does that imply it is provable?

Let’s say a conjecture suggests for all $x$ in an infinite domain, $P(x)$ is true(conjectures such as the Goldbach Conjecture, the Collatz Conjecture, etc). Now, could we say no such conjecture could ...
416 views

60 views

### Searching for a conjecture that is true until the 127 power of n. [duplicate]

I am searching for a mathematical conjecture that is true until something like n = 117, or n = 127, or a number close. It is an ...
69 views

### A conjectured continued fraction involving non-polynomial patterns

After having been devoting some time for many years to experimental mathematics, I am thinking to publish the details of some of my most fruitful computing workflows for discovering identities. I ...
158 views

### Conjecture: No positive integer can be written as $x^y+y^x$ in more than one way

Today, I came up with the following problem when trying to solve this. Are there distinct integers $a,b,x,y>1$ such that the equation $$a^b+b^a=x^y+y^x$$ holds? That is, Is there ever an integer ...
68 views

181 views

161 views

### Infinitude of super happy primes

Similar to happy primes, I define super happy primes by the following process: $(1)$ Find the sum of the digits raised to the power of themselves. Ex. $13$ gives sum $= 1^1 + 3^3 = 28$ $(2)$ If ...
117 views

### Is there a maximum number of consecutive sexy prime pair sums that are all divisible by $10$?

I was finding the sums of pairs of sexy primes (prime numbers that differ by 6) and noticed that there are a lot of pairs who's sum is divisible by $10$. Ex. $(7, 13)$ as $7+13=20$ and $20$ is ...
### Conjecture: all complex roots of $\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}$ are real
Conjecture: $$\left[n\in\mathbb{Z}^+,z\in\mathbb{C},0=\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}\right]\Rightarrow z\in\mathbb{R}$$ This conjecture has been verified for $n\in\{1,2,4\}$. The ...