Questions tagged [conjectures]

for question related to conjectures.

Filter by
Sorted by
Tagged with
7 votes
2 answers
2k views

Is there any progress toward solving Gilbreath's conjecture?

Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the ...
10 votes
2 answers
983 views

Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange. Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $ And let $ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $ It appears that we have $$\sup ...
1 vote
0 answers
138 views

Optimal covering trails for every $k$-dimensional cubic lattice $\mathbb{N}^k := \{(x_1, x_2, \dots, x_n) : x_i \in \mathbb{N} \wedge n \geq 3\}$

After a dozen years spent investigating this particular class of problems, I finally give up and I wish to ask you if any improvement is achievable from here on. The general problem is as follows: Let ...
3 votes
4 answers
1k views

A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.) Let $\sigma(z)$ denote the sum of ...
7 votes
0 answers
580 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second Hardy-Littlewood ...
3 votes
0 answers
211 views

Uses of excluded middle on a conjecture that can be rewritten constructively with this trick

An interesting proof technique is to use the law of excluded middle on a conjecture. There are proofs using LEM on the Riemann hypothesis for example. Constructively this is disallowed (if you can ...
8 votes
0 answers
431 views

Does the interior of Pascal's triangle contain three consecutive integers?

This question defeated Math SE, so I am posting it here. Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. ...
7 votes
1 answer
493 views

About semiprimal representations of $1$

Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...
20 votes
1 answer
906 views

Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here. Is the following conjecture true or false: Given any five coplanar points, we can always draw at least ...
1 vote
0 answers
155 views

Initial conditions to falsify Rowland's conjecture

Based on the Rowland's paper (A natural prime-generating recurrence), is there any theorem to show that for which initial condition $a(1) = k$ the conjecture can be falsified? For example, for $k$ ...
3 votes
1 answer
799 views

What do we know about Lucky numbers?

I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics. Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
6 votes
1 answer
319 views

Different flavours of Vassiliev Conjecture

There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
11 votes
1 answer
840 views

Seeking proof of the Cuckoo Cycle Conjecture

Allow me to introduce the Cuckoo Cycle Proof of Work, and conclude with a closely related Conjecture about random graphs, which I hope someone can find a proof for. Cuckoo Cycle is named after the ...
2 votes
2 answers
474 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
12 votes
2 answers
516 views

Sylvester–Gallai theorem with circle version, plane version and curve version?

The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either All the points are collinear; or There is a line which contains exactly two of the ...
3 votes
0 answers
355 views

A conjecture on consistent monotone sequences of polynomials in Bernstein form

A Conjecture In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
2 votes
0 answers
104 views

On a subset of the $abc$ triples

The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and $$c > \...
1 vote
0 answers
312 views

How would one go about solving this conjecture concerning exponential Diophantine equations?

I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...
15 votes
3 answers
600 views

Maximum matching in a graph with no "shortcuts"

For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^...
2 votes
0 answers
261 views

Conjecture about primes and Fibonacci numbers

I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's ever possible to find a ...
13 votes
1 answer
13k views

The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
10 votes
1 answer
2k views

Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
1 vote
1 answer
180 views

Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers

Let $k \in \mathbb{Z}^+$. Is it possible to prove that, for some given $m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$, there are only finitely many $k$ such that the closed ...
15 votes
0 answers
336 views

Do primes of the form $4k+1$ ever lead the greatest prime factor race?

Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
0 votes
0 answers
117 views

Conjecture on the unsolvability of the $\{3 \times 3 \times \cdots \times 3\} \subseteq \mathbb{R}^k$ dots problem starting from the central point

In 2020 (see Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm, JFMA, 3(2), p. 96), I conjectured that, in the Euclidean space $\mathbb{R}^k$, we can cover any given set of ...
-1 votes
1 answer
177 views

Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$

I try here because I expect I cannot have any answer on MSE : Problem : Let : $$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$ Then it seems $\exists y\in(0,1)$ and $...
17 votes
5 answers
2k views

Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$

in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?
9 votes
2 answers
571 views

Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to: \begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
1 vote
0 answers
144 views

Stronger conjectured inequality for area of a polygon

Four years ago, I proposed an inequality related to area and sides of a polygon. After computer checking, I conjecture that the previous inequality can be strengthened as follows: Let $A_1A_2\cdots ...
0 votes
0 answers
82 views

Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$

Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia) $$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...
2 votes
0 answers
285 views

How soon can we represent a number as the sum of two primes?

Posting in MO since it was unanswered in MSE. Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
6 votes
1 answer
1k views

Is there any relationship between Szemerédi's theorem and Sunflower conjecture?

I have observed some similar things between a reformulation of the Sunflower conjecture (see also conjecture 1.3 in Improved bounds for the sunflower lemma) and Szemerédi's theorem such that for ...
12 votes
2 answers
902 views

An analogue of the Bass-Quillen conjecture with power or Laurent series

The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
1 vote
0 answers
2k views

Collatz conjecture in all its variants

There are all kinds of execution variants to the collatz conjecture for when hitting an odd number: $3n+1$ or $3n+3^a$ or $1.5n + 0.5$ or $1.5n + 1.5$... . The assumption is: proving any of them will ...
15 votes
1 answer
1k views

Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern. We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
2 votes
1 answer
296 views

Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
20 votes
1 answer
586 views

Distinct exponents in the factorization of the factorial, a problem of Erdős

In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...
7 votes
3 answers
1k views

Prime constellation conjectures

This is a simple question about terminology and provenance. I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture. I've encountered Polignac's ...
2 votes
0 answers
257 views

A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?

The definitions are from these two questions: https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers https://math.stackexchange.com/questions/4349186/trying-to-understand-...
11 votes
3 answers
657 views

An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
23 votes
1 answer
3k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
8 votes
2 answers
2k views

Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple. I have often heard of the statement that the SZC is stronger than the Riemann ...
78 votes
9 answers
9k views

Mathematical conjectures on which applications depend

What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
4 votes
1 answer
233 views

Total coloring conjecture for Cayley graphs

The total coloring conjecture (TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree ...
10 votes
1 answer
457 views

A counterexample to a conjecture of Lawson

Yau quotes Lawson as having formulated the following conjecture [1]: Let $M$ be an embedded minimal surface in $\mathbf{S}^3$. Prove that the two domains in $\mathbf{S}^3$ divided by $M$ have equal ...
5 votes
1 answer
393 views

Is there an even number $a$ such that $a^{2^{n}}+1$ is prime for infinitely many $n$?

Is there an even number $a$ such that $\{n: a^{2^{n}}+1 \text{ is prime} \}$ is an infinite set? Let $a$ be even. Is there infinitely many $n$ such that $a^{2^{n}}+1$ is composite?
0 votes
0 answers
96 views

A special case of Frankl's conjecture. A question about known results

Let's recall a Frankl's conjecture. Consider a finite family of finite sets $\mathcal{F}$, such for every pair of sets $A\in \mathcal{F}$ and $B\in\mathcal{F}$, we have $A\cup B\in\mathcal{F}$(the ...
5 votes
0 answers
313 views

Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures

I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...
6 votes
1 answer
279 views

Lemoine-Lozada circles

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
-1 votes
1 answer
510 views

Questions about the abc conjecture [closed]

Question. Is there an integer $n_0 \geq 2$ such that $$\left\{\frac{c}{rad(abc)^{n_0}}: a, b >0,\; c=a+b,\; \gcd(a, b)=1\right \}$$ is bounded? The abc conjecture can directly deduce this ...