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

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53 views

Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
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1answer
61 views

A conjecture about binary palindromes and arithmetic derivatives

Corrected question. From the sequence of binary palindromes A006995 (eg. 1001001001001) the sequence of possible gaps between consecutive palindromes contain the elements: ...
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1answer
354 views

Continued fraction for Apéry's constant conjectured by The Ramanujan Machine

Recently the following identity was conjectured by The Ramanujan Machine: $$ \frac{8}{7\zeta(3)}=1-\frac{u_1}{v_1-\frac{u_2}{v_2-\frac{u_3}{v_3-\ddots}}}, $$ where $u_n=n^6$ and $v_n=(2n+1)(3n^2+3n+1)...
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Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?

Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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Term for example that is counterexample for “Conjecture A AND Conjecture B”

I am looking at something where there are two conjectures, Conjecture A and Conjecture B. I have Example X. It isn't a counterexample for Conjecture A or for Conjecture B but it is a counterexample ...
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1answer
49 views

A proof (?) for $k = 1 \implies q \neq 5$, if $q^k n^2$ is an odd perfect number with special prime $q$

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form (i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$). Inspired by mathlove's answer to ...
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1answer
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Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?

CONTEXT This question is a result of considerations stemming from this closely related MO question. INITIAL QUESTION My question is as is in the title: Is there an odd $x$ such that $$2x^2 \equiv 0 \...
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Addition to previously asked question on Generalized Carmichael Numbers

I had previously asked a question on mathstack exchange (Conjecture on The Generalized Carmichael Numbers) concerning with a conjecture I had discovered. I worked on the problem for a long time and ...
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Real world applications of Schanuel's Conjecture

I'm doing my senior capstone on Schanuel's conjecture and in my final presentation I wanted to discuss why this conjecture is important. I have found tons of applications in field theory and proving ...
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Asymptotic series for the inverse Riemann ζ-function

Consider the positive branch of the Riemann $\zeta$-function: $\zeta:(1,\infty)\to(1,\infty)$ and its inverse $\zeta^{\small(-1)}:(1,\infty)\to(1,\infty)$ satisfying $\zeta\left(\zeta^{\small(-1)}(x)\...
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Is there any algorithm generating “all” chrodal graphs of order n?

I defined a procedure $\Gamma$ to generate graphs following some rules. The obtained graphs form a class of graphs $\mathcal{G}$. I found out that these graphs are chrodal. In the other hand, I ...
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Progress on a conjecture of Burnside…

Given a group $G $, the set of automorphisms of $G $ also forms a group, $\rm {Aut}(G) $,with composition as the operation. An inner automorphism is one determined by conjugation by some element $g\in ...
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Proof of conjecture about orthogonalized Specht polynomials.

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a $r$-part partition of integer $N$ ($\lambda\vdash N$), i.e., $$\sum_{i=1}^r{\lambda_i}=N,$$ such that $\lambda_i\leq\lambda_j$ for $r\geq j>...
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1answer
68 views

Is there a way to prove that $A^3 - (A-1)^3 = 13^x$ is false when $x$ is an odd positive integer greater than $1$?

In the case of $A^3 - (A-1)^3 = B^x$ we can find some rare examples, such as: $8^3-7^3 =13^2$ $28712305723921^3−28712305723920^3=49731172316281^2$ But according to the Beal's Conjecture $A^3 - (A-1)^3 ...
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Are there any other examples besides $8^3-7^3 =13^2$ for $C^z -(C-1)^z=A^2$ where $C$ and $z$ are positive integers greater than $1$ and $z$ is odd?

I was trying to study the Beal's Conjecture, which states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must ...
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1answer
112 views

On a conjectured upper bound for $k=\nu_q(N)$, if $N=q^k n^2$ is an odd perfect number with special prime $q$

Let $N=q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ as $I(x)=\sigma(x)/x$ ...
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185 views

A Conjecture on the Existence of Integer Solutions for $(f_1)^2 + (f_2)^2 +( f_3)^2… = I^2$

Using fractions($f_n$) where integer $n$ is the number of fractions, prove that the sum of the squares $f_1$ to $f_n$ has no integer($I$) solutions when $n >= 1$ given each fraction has a distinct ...
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A possible solution for a specific case of the Beal's Conjecture?

According to wikipedia's page on Beal's Conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ ...
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A Question on Integers and Fractions

Given that $a,b,c,d,e,f$ are all distinct positive integers, prove that there are no solutions for the following knowing their fractions are non-integers and completely simplified: $$\left(\frac{a}{c}\...
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A possible approach for the Beal's Conjecture?

Beal's conjecture is a generalization of Fermat's Last Theorem. It states: If $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $...
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Product of Consecutive integers problem [duplicate]

Product of Consecutive integers problem I've begun pondering the following problem, and have found myself unable to advance on it; it goes like this: Prove that the product of three consecutive ...
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Beal's Conjecture can the following specific case be proved?

According to wikipedia's page on Beal's Conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ ...
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1answer
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A problem with the equivalent form of Frankl conjecture.

The Union Closed Sets Conjecture (or Frankl conjecture) is described in this link: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture. It is known that this problem has an equivalent form in ...
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Collatz conjecture with limited divisions

The Collatz conjecture defines sequence (where $v_n$ is 2-adic valuation of $a_n$) $$a_{n+1}=3\left(\frac{a_{n}}{2^{v_n}}\right)+1.$$ The conjecture is that for every $a_0\in\mathbb N$, the sequence $...
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115 views

Is this Pythagorean triple formula original?

I am hesitant because I have asked this question before in a different form here. In 2009, I knew nothing about Pythagorean triples, not even Euclid's formula. In my ignorance, I did a brute force ...
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Differential Geometry Inequality Conjecture?

Question and Conjecture In a Lorentzian metric (Only one time dimension) of signature: $(+,-,-,-,\dots)$ Lets say I have a line element $ds^2 = \sum_{\mu , \nu = 0}^n g^{\mu \nu} d \tilde x_\mu d \...
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2answers
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Yet another Collatz generalization and conjecture

Observe that Collatz (the $\frac{3n+1}{2}$ version) can be written as: $$n_{i+1}:= \begin{cases}n_i+\left\lceil\frac{n_i}{2}\right\rceil\quad\text{ if odd }n_i \\ n_i-\left\lceil\frac{n_i}{2}\right\...
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1answer
74 views

Is the conjecture true? $3^n - 2^m = 1$ has infinitely many solutions, where n, m are natural numbers.

Is the conjecture true? $3^n - 2^m = 1 $ has infinitely many solutions, where n, m are natural numbers. More generally, $P^n-Q^m=1$ has infinitely many solution for n, m ϵ {1,2,3,…} where P is odd, Q ...
2
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1answer
76 views

Prime counting function in large-ish intervals--two questions/conjectures

Consider two positive integers $2 < a < x$. The following two statements would seem to follow intuitively from the PNT, but I'm wondering if either has been proven or discussed. Here, $\pi(n)$ ...
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209 views

Conjecture of Differential Geometry Line Element identity?

Question Lets say I have a line element $ds^2 = \sum_{\mu , \nu = 0}^n g^{\mu \nu} d \tilde x_\mu d \tilde x_\nu $ and $1.$ It is of the form $ds^2 = \sum_{\mu=0}^n g^{\mu \mu} d \tilde x_\mu d \...
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Conjecture: For almost all $n$, the sum of $n \bmod k$ for $k<n$ is unique.

It looks to me like $$f:n\mapsto \sum_{k=2}^{n-1}{(n\bmod k)}$$ is unique to $n$, with the only exceptions being $2^m-1$ and $2^m$, which share a value. In other words, the set $\{f(n): n\in\mathbb N\...
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There are $2^{\omega(a)}$ pairs $(b,c)$ such that $a=bc$ and $\gcd(b,c)=1$

It looks as there are $2^{\omega(a)}$ ordered pairs $(b,c)$ such that $a=bc$ and $\gcd(b,c)=1$, where $\omega(a)$ is the number of different prime factors of $a$. Proof? It's also true that there are ...
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152 views

Conjecture: if collatz($n$) reaches 1, it must do so in at most $2^{n+1}$ steps

Let $f(n)$ give the number of Collatz iterations it takes to yield a number less than $n$, using the "shortcut" version of Collatz which applies $n\rightarrow (3n+1)/2$ for odds. Suppose it ...
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In regards to the Beal's conjecture

Studying the Beal's conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a common ...
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1answer
54 views

Conjecture: primes non-decreasing from $(n,2n)$ to $(2n,4n)$.

More generally, I assert: $$\pi(na^k)-\pi(na^{k-1})\leq \pi(na^{k+1})-\pi(na^k)\quad\text{for }n,a,k \in \mathbb N.$$ The most useful case of this is probably to say that if there are $m$ primes in $[...
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1answer
61 views

Let $n=am+r$ with $m \ge a>5$ and $r \ge 0$. Prove that $\phi(n) \ge m$

I understand that for all $n>6$, we have $\phi(n) \ \ge \ \sqrt n $. This fact however is not getting me anywhere towards the proof of the claim in question. ($a, m, r$ are all integers)
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358 views

Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ is an odd perfect number?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the abundancy index $$I(x)=\frac{\sigma(x)}{x}$$ where $\sigma(x)$ ...
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1answer
60 views

Can you spot the mistake here? [closed]

assume that : $$x^{3} + y^{3} = z^{3}$$ so : $$z^{3} - y^{3} = x^{3}$$ $$z^{3} ≡ y^{3}\bmod x$$ $$z ≡ y\bmod x $$ $$z-y = x $$ and now let's plug this result to the original equation: $$(z-y)^{3} + y^{...
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On Andrica's conjecture.

We have Andirca's conjecture as $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$. I have no clue if this is a proof. I was just messing around. (And plus, my methods are probably too elementary). Denote $g(n)$ as ...
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1answer
56 views

Integer solutions to $r\cdot a^2+t=s\cdot b^2$ forms sequences with $a_{n+1}\cdot b_n-a_n\cdot b_{n+1}$ is invariant

Conjecture: Given natural numbers $r,s,t$. Suppose there are an infinite number of solutions $a_n,b_n \in \mathbb N$ to $ra_n^2+t=sb_n^2$, where $a_n, b_n$ correspond to increasing sequences of ...
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A curious infinite product of factorials

I found the following infinite product of factorials without proof: $$\small\prod_{n=1}^\infty\frac{{(2 n)!}^{20}\,{(8 n)!}^{32}\,{(32 n)!}^2}{{n!}^4\,{(4 n)!}^{37}\,{(16 n)!}^{13}}=\\\frac{\sin ^{14}\...
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1answer
226 views

Sums of integer powers similar to Prouhet–Tarry–Escott problem

Recall Prouhet–Tarry–Escott problem. Its solution shows that certain sums of powers of integers can be made to vanish simultaneously if their signs are chosen to follow the Thue–Morse sequence, e.g. $$...
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1answer
164 views

Conjecture regarding the sum of prime factors of $x!$ and $(x-1)!$?

I think using dodgy means I can show the following: Let $\lambda(x)$ be the sum of primes in $x$. For example: $$ \lambda(2) = 2$$ $$ \lambda(4) = 2 + 2 = 4$$ $$ \lambda(6) = 3 + 2 = 5$$ Then for ...
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2answers
96 views

Does the Bunyakovsky conjecture apply for every cyclotomic polynomial?

The Bunyakovsky conjecture states that for every irreducible polynomial $\ f\in \mathbb Z[x]\ $ with positive leading coefficient, there are infinite many primes of the form $\ f(m)\ $ , where $\ m\ $ ...
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63 views

Can you prove this number theory conjecture?

I think I can prove the following (using a conjecture): $$ 2n (\sum_{i=2}^n p_i ) \geq p_n(p_n+1)/2 - 6$$ where $p_i$ is the $i$th prime. Can one prove this without any conjecture?
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Conjecture : A lower bound for the prime counting function .

Well it take my a little bit of time to find it but now I think that my conjecture is ready . Conjecture : Let $n\geq 100$ and then define the sum : $$S(n)=\sum_{k=1}^{n}\frac{1}{\operatorname{...
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76 views

Is there always a prime $p$ which falls short of $2^n$ by at most a square-root factor?

For fixed $n > 0$, does there always exist a prime $p$ such that $p < 2^n \leq \lceil p + \sqrt{p} \rceil$? Would this follow from any known conjectures? This follows immediately from Opperman's ...
3
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1answer
101 views

Bézout giving some spare coins to Frobenius? (the Coin Problem)

Let $a$ and $b$ be relatively prime positive integers and write the two minimal Bézout's identity pairs, $\tag 1 (s)\,a + (t)\,b = 1$ $\tag 2 (u)\,a + \;(v)\,b = 1$ where the extended Euclidean ...
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1answer
62 views

Is it true that every closed formula is decidable? Why?

I was wondering if every closed formula is decidable (in a complete system). Clearly a non closed formula is not decidable, since it can take some values for which it is true, and other for which it ...
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1answer
76 views

Conjecture: any factorial is a sum of two Jordan-Polya numbers.

A Jordan-Polya number A001013 is a product of factorials $\prod^m_{k=1}a_k!$. For $n=2,\dots,12$ there are Jordan-Polya numbers $a,b$ such that $n!=a+b$. ...

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