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

In mathematics, a conjecture is a conclusion or a proposition based on incomplete information, for which no proof has yet been found. There are many famous conjectures. Some of them have been proved, in which case they're technically no longer conjectures, and some have been disproved, and many others which still remain open, are yet to be proved or disproved. A few well-known conjectures are famous enough to have their own tag on this site: the Collatz conjecture, the ABC conjecture, and Goldbach's conjecture.

A list of some well-known conjectures can be found on Wikipedia.

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A Conjecture on The Generalization of Quadratic Reciprocity Law

Is there any way to prove the following conjecture regarding the Generalization of Quadratic Reciprocity Law. The statement being, $$ \left(\dfrac{a_1}{a_2}\right)\left(\dfrac{a_2}{a_3}\right)…
user170039
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Lines formed from vertices of n-gons equate to triangular numbers.

Noticed something neat tonight! The number of unique lines you can form by connecting the vertices of an n-gon is equal to the (n-1)th triangular number. (e.g. in a square all 4 veritices make 4 lines (4 sides to a square), and then I can also…
Albert Renshaw
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Is the following infinite product of fractions of linear factors equal to an exponential function or not?

Is the following infinite product: $$ \prod_{\substack{(a,b) \in \mathbb{Z}^2 \\ a > b}} \frac{x+b}{x+a} $$ defined? If so, does it simplify to an exponential function of $x$? The subscript is over all integer pairs $a$ and $b$ such that $a>b…
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some property of the simple continued fraction for the riemann-zeta function

Define the well known Riemann-zeta function $$\zeta(n)=\sum_{k=1}^{\infty}\frac{1}{k^n}$$ for natural number $n\gt1$ and let $a_1(n)$ be a sequence $a_1(1),a_1(2),a_1(3),\dots$ of the first partial quotients of the simple continued fraction for…
Nicco
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Two different polynomial expressions for certain primes

Here is an other random conjecture which I have no clue how to prove: $a,b\in\mathbb N^+\wedge a^2+b^2+ab\in\mathbb P\implies\exists$ $A,B\in\mathbb N^+:A^2+B^2-AB=a^2+b^2+ab$. Tested for $a,b<20,000$ on my 32 bit tabloid. I would like to see a…
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What must the value of $y$ be so that $a$ and $b$ are integers?

I have made some sort of conjecture that in the equation: $$a^2 = b^2 + 2y^2.$$ $y$ must equal $(2n + 1)\sqrt {8}$ such that $n \in \mathbb{W}$ for $a$ and $b$ to be integers, or at least Pythagorean Triples. All I believe that is most correct is…
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Goldbach Conjecture and Conjecture of Preservation of Nature of Numbers

About Goldbach Conjecture: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture Doubt 1 "The best known result is due to Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes.". About this…
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$P^2 + P + 1$ is prime if $P$ is prime.

I just thought of it. I don't know if there is one such conjecture or a proven problem or if this is a new conjecture. If it already exists where can I find this problem. And most importantly how to prove it??
jnyan
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Can you spot the mistake here?

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^{3} =…
PNT
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Every 'decreasing' or 'increasing' infinite sequence whose sum converges contains at least one term of magnitude 0

Note on the title: This conjecture is not restricted to real numbers, 'decreasing' and 'increasing' were used because of the character limit. The actual conjecture is that every sequence whose terms strictly increase or decrease in magnitude…
R. Burton
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$k$-tuple conjecture.

This conjecture is false. See this post Time is running out Suggest notation for Steps 1 and 2. Earn the bonus. For each $k\in\mathbb{Z^{+}}$. Step 1: Create a list $(1,1,1,1...,1)$ of length $k^2+2k$. Step 2: For all n: $1< n \leq \pi(k)$, at index…
Fred Kline
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Why is that a conjecture is written and its proof can not be done for years?

Why is it that if a conjecture is written and its proof can not be done for years, means if its statement is understandable in language, why is the proof not possible if the conjecture is correct? E.g., the proof of Fermat's last theorem (which was…
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