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

250 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
47
votes
0answers
653 views

Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
15
votes
0answers
240 views

Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
14
votes
1answer
570 views

$\sum\limits_i a_i^2\sum\limits_i b_i^2+\left(\sum\limits_ia_i b_i\right)^2\geq \sqrt{\sum\limits_i a_i^4\sum\limits_i b_i^4}+\sum\limits_ia_i^2b_i^2$

I have no idea about how to prove (or disprove) the following inequality: $$ \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\geq \sqrt{\left(\sum_{i=1}...
13
votes
0answers
151 views

Rare gaps between integers having the same number of co-primes

Let $\varphi(x)$ be the Euler totient function. For $k = 1,2,3,\ldots$ I calculated the number of solutions of $\varphi(x) = \varphi(x+k)$. I observed that we have very few solutions when $k = 3,9,15,...
13
votes
0answers
343 views

Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
12
votes
0answers
400 views

Is $\lfloor \zeta(-n) \rfloor$ only prime for $n=23$?

I searched for primes of the form of $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, for a range of $n \le 10^4$ on PARI/GP and found $\lfloor \zeta(-n) \rfloor$ is only prime for $n=23$. My ...
12
votes
0answers
170 views

Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the complete elliptic integral of the first kind $K(k_\color{blue}m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\...
11
votes
0answers
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 ...
10
votes
0answers
311 views

A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\...
9
votes
0answers
74 views

Conjectures for which there are strong heuristic arguments both for and against

I find the conjecture that there are infinitely many Fermat primes to be very interesting, because there is a very reasonable-seeming and semi-quantitative heuristic argument for the conjecture, but ...
9
votes
0answers
146 views

Complete this table of general formulas for algebraic numbers $u,v$ and $_2F_1\big(a,b;c;u) =v $?

(This extends this post.) Given fixed rationals $a,b,c,$ the problem of determining, $$_2F_1\big(a,b;c;u) =v $$ such that both $u,v$ are algebraic numbers may be solved by appealing to modular ...
9
votes
1answer
160 views

Question regarding the status of Erdős' conjectures

Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?
9
votes
0answers
553 views

Asymptotic FLT $\implies$FLT using ABC Conjecture

Edit: I'm beginning to suspect I either misread the sources or perhaps something wasn't stated, but it's my guess now that there is no nontrivial way of showing the ABC conjecture implies Fermat's ...
8
votes
0answers
72 views

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

While discussing prime powers and divisors, I came up with the following problem. Examples $\to$ prime $p=3$ digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,. $\to$ prime $p=7$ ...
7
votes
2answers
89 views

For all $n\in\mathbb{N}$ with $n>1$ there exists an $m\in\mathbb{N}$ with $m<n$ such that $mn+1$ is prime.

I was using Python to explore some interesting lexicographically earliest sequences. I was looking into the sequence of numbers such that $a(1)=1$ and $a(n)$ is the smallest number such that $a(k)n+1$ ...
7
votes
0answers
165 views

A generalization of the abc-conjecture?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
7
votes
0answers
119 views

Conjecture: There is no $n\neq3$ such that $\arctan(1)+\arctan(2)+\cdots+\arctan(n)=\frac{k\pi}{2}$, with $k$ integer.

Fact: $\arctan (1) + \arctan (2) + \arctan (3) = \pi$. Conjecture: there is no $n \neq 3$ such that $\arctan (1) + \arctan (2) + \cdots + \arctan (n) = \frac{k\pi}{2}$, with $k$ integer. I know a ...
7
votes
0answers
224 views

Pairs of palindromic primes without $1$ and have a palindromic product

While discussing about prime numbers with other users, I noticed that: $(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are ...
7
votes
0answers
372 views

Yet an other conjecture about odd numbers $n=a+b$ such that $a^2+b^2$ is prime

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding ...
7
votes
0answers
159 views

Can we prove this conjecture concerning Pell equations?

For every positive integer $n$, not being a perfect square, denote the fundamental solution of the Pell equation $$x^2-ny^2=1$$ with $(a,b)$. In other words, $b$ is the smallest positive integer such ...
7
votes
0answers
132 views

The $\gcd$ operator commutes with functions defined by linear recurrence relations

Given $a,b\in\mathbb Z^+$, and let $F_{a,b}:\mathbb N\to\mathbb N$ be a function such that $F_{a,b}(0)=0$ and $F_{a,b}(n+1)=a\cdot F_{a,b}(n)+b\cdot F_{a,b}(n-1)$. $F_{1,1}$ correspond to the ...
7
votes
0answers
280 views

Consequences of a proof of Schanuel's conjecture?

Schanuel's conjecture is a strengthening of the Lindemann-Weierstrass-threorem. It states If $\lambda_1,\cdots \lambda_n$ are complex numbers linear independent over $\mathbb Q$, then $$\mathbb Q(\...
7
votes
0answers
368 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
6
votes
0answers
141 views

Conjectures on the primality of $\sum\limits_{i=1}^n p_i^{p_i}$ and $\sum\limits_{i=1}^n (-1)^ip_i^{p_i}$

Inspired by Is $29$ the only prime of the form $p^p+2$?. Claims on prime powers and their alternating sums Consider the expressions $\mathcal P=\sum\limits_{i=1}^n p_i^{p_i}$ and $\mathcal Q=\sum\...
6
votes
0answers
201 views

Are $3$, $9$ the only natural numbers $n$ for which both $2^n-n$ and $2^n+n$ are prime?

I searched for natural numbers $n$, where $2^n-n$ and $2^n+n$ are both prime for the range of $n \le 10^5$ on PARI/GP and found that 3, 9 are the only solutions in this range. Note that since $2^n-n$...
6
votes
0answers
72 views

New results on Identical Binomial Coefficient?

Are there any nontrivial identical binomial coefficients found other than: $$ {16 \choose 2}={10 \choose 3}=120 \\ {21 \choose 2}={10 \choose 4}=210 \\ {56 \choose 2}={22 \choose 3}=1540 \\ {120 \...
6
votes
0answers
152 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
6
votes
0answers
98 views

The fractal dimension of the Kolakoski sequence is $2-1/e$

The Kolakoski sequence, which is defined as the infinite sequence of symbols {1,2} that is its own run-length encoding (Wikipedia), has been suggested to be self-similar$^{1}$. The fractral ...
6
votes
0answers
72 views

There exists infinitely many $N$ such that $\{\sum_{n=2}^N\log(n)\}<\varepsilon$

I am wondering whether or not the following result is true: For all $\varepsilon>0$, there exists infinitely many $N\in \mathbb N$ such that $$S_N:=\left\{\sum_{n=2}^N\log(n)\right\}<\...
6
votes
0answers
146 views

Help finding the flaw in this proof

Today I spent my afternoon trying to understand why the Hardy-Littlewood's Second Conjecture is considered as a problem of such a great difficulty. I got a fairly strange result, so I decided to post ...
6
votes
0answers
104 views

How close are we to knowing the rate of convergence to $0$ of $\prod_{p\le x}(1-1/p)^{-1}-e^\gamma\log x $?

This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic. Despite what one can read on the MathWorld page about Mertens' third ...
6
votes
0answers
171 views

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
6
votes
0answers
287 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
6
votes
1answer
610 views

Improvement on the concept of separating families for the union-closed sets conjecture?

The union-closed sets conjecture states the following. Let $F$ be a finite family of finite sets that is union-closed and let $\cup (F)$ be the union of all sets in $F$. Then we can find an element in ...
5
votes
0answers
259 views

For any $k \gt 3$, if $n!+k$ is a perfect power then does there exist any $n\gt k$?

A while ago, I asked a similar question For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true? where users mathworker21 and WE Tutorial School proved that for non-square $k$, $n\...
5
votes
0answers
165 views

Is this elementary-number-theoretic conjecture already mentioned somewhere or it is entirely new?

While I was doing some research in elementary number theory I discovered some regularities that seem to be very promising. First, function $d$ can be defined as $d(n)=d(\prod_{i=1}^{k(n)} p_i^{r_i})=...
5
votes
0answers
73 views

Every number is a sum of (exactly/at most) some (distinct or not) primes

Introduction I have observed some heuristic results, and also attempted to formulate a proof, of the conjecture I formulated based on those heuristics. But first, I want to ask about relevant ...
5
votes
1answer
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 ...
5
votes
0answers
110 views

Conjecture about primes and the greatest common divisor

Conjecture: Given $m,n\in\mathbb N^+$, one odd and one even, there are two primes $p,q$ such that $|mp-nq|=\gcd(m,n)$. I hope MSE can determine its validity. From time to time, when testing my ...
5
votes
0answers
84 views

“Schäffer's conjecture” on equation $1^k+2^k+\cdots+x^k=y^n$

In 1956 J. J. Schäffer proved that the equation $$1^k+2^k+\cdots+x^k=y^n$$ for fixed integers $k\ge1,n\ge2$ has only finitely many solutions in positive integers unless $(k,n)\in\{(1,2),(3,2),(3,4),(...
5
votes
0answers
165 views

Show $\lim_{n\to \infty}{|A_{n}|+|A_{n-2}| \above 1.5pt |A_{n-1}|}=2$

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a prime; otherwise equals to $0$. I write $|A(n)|$ to count the number of $1$'s in $A(n)$. Is it ...
5
votes
0answers
116 views

Are there infinitely many primes in any sequence determined by a $k$ that is not a Sierpinski number?

Consider the sequence of numbers ranging over $n$ of the form $k\cdot a^n + 1$ for a fixed, odd natural number $k$. The number $k$ is considered a Sierpinski number if the sequence determined by that ...
5
votes
1answer
186 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
5
votes
1answer
667 views

Totient summatory function

Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a ...
5
votes
0answers
321 views

Has every finite group a minimal presentation?

For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with ...
5
votes
1answer
215 views

A conjecture on the closeness of twin primes

Let $p_1$ and $p_2$ be twin primes, and let $p_1-1=a_1\times b_1$ and $p_2+1=a_2\times b_2$ be such that $|b_1-a_1|$ and $|b_2-a_2|$ are minimised. Similarly, let $p_1+1=p_2-1=a\times b$ be such that ...
4
votes
0answers
71 views

To prove the existence of solution(s) of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$

To prove if a certain equation has a solution, we do not need necessarily solve that equation. Example: Is there any point of the curve $y=\sin^2(x)/x$ between $x=0$ and $x=\pi$ so that the slope of ...
4
votes
0answers
62 views

Grimm's Weak Conjecture

On Wikipedia it says that the following weaker version of Grimm's conjecture is also still open: "A product of $k$ consecutive composite numbers has $k$ pairwise distinct prime factors." Is it true, ...
4
votes
0answers
134 views

Conjecture about Jensen's inequality and polynomials

Hi it's related to the following conjecture An inequality for polynomials with positives coefficients : We have the first conjecture : Let $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+...
4
votes
0answers
93 views

Minimal counterexample to $\frac{p - 1}{p^2}$-conjecture

There existed once a “folklore” conjecture that stated: Suppose $p$ is a prime. Then any finite group $G$ with $> (1 - \frac{p-1}{p^2})|G|$ elements of order $p$ has exponent $p$ This ...

1
2 3 4 5