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Alexa's Conjecture on Primality ★★

Author(s): Alexa

Definition   Let $ r_i $ be the unique integer (with respect to a fixed $ p\in\mathbb{N} $) such that

$$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i < p. $$

Conjecture   A natural number $ p \ge 8 $ is a prime iff $$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$

Keywords: primality

Diophantine quintuple conjecture ★★

Author(s):

Definition   A set of m positive integers $ \{a_1, a_2, \dots, a_m\} $ is called a Diophantine $ m $-tuple if $ a_i\cdot a_j + 1 $ is a perfect square for all $ 1 \leq i < j \leq m $.
Conjecture  (1)   Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

Conjecture  (2)   If $ \{a, b, c, d\} $ is a Diophantine quadruple and $ d > \max \{a, b, c\} $, then $ d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}. $

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Are there only finite Fermat Primes? ★★★

Author(s):

Conjecture   A Fermat prime is a Fermat number \[ F_n  = 2^{2^n }  + 1 \] that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.

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Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (re-designed) ★★

Author(s):

Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (re-designed)

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Square achievement game on an n x n grid ★★

Author(s): Erickson

Problem   Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $ n \times n $ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.

Keywords: game

New War Dragons Free Rubies Cheats 2024 Tested (extra) ★★

Author(s):

New War Dragons Free Rubies Cheats 2024 Tested (extra)

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Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

Olson's Conjecture ★★

Author(s): Olson

Conjecture   If $ a_1,a_2,\ldots,a_{2n-1} $ is a sequence of elements from a multiplicative group of order $ n $, then there exist $ 1 \le j_1 < j_2 \ldots < j_n \le 2n-1 $ so that $ \prod_{i=1}^n a_{j_i} = 1 $.

Keywords: zero sum

Erdös-Szekeres conjecture ★★★

Author(s): Erdos; Szekeres

Conjecture   Every set of $ 2^{n-2} + 1 $ points in the plane in general position contains a subset of $ n $ points which form a convex $ n $-gon.

Keywords: combinatorial geometry; Convex Polygons; ramsey theory

List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

Conjecture   Every $ K_t $-minor-free graph is $ c t $-list-colourable for some constant $ c\geq1 $.

Keywords: Hadwiger conjecture; list colouring; minors

Order-invariant queries ★★

Author(s): Segoufin

Question  
    \item Does $ {<}\text{-invariant\:FO} = \text{FO} $ hold over graphs of bounded tree-width? \item Is $ {<}\text{-invariant\:FO} $ included in $ \text{MSO} $ over graphs? \item Does $ {<}\text{-invariant\:FO} $ have a 0-1 law? \item Are properties of $ {<}\text{-invariant\:FO} $ Hanf-local? \item Is there a logic (with an effective syntax) that captures $ {<}\text{-invariant\:FO} $?

Keywords: Effective syntax; FMT12-LesHouches; Locality; MSO; Order invariance

Bingo Blitz Cheats Generator Unlimited No Jailbreak (Premium) ★★

Author(s):

Bingo Blitz Cheats Generator Unlimited No Jailbreak (Premium)

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Is Skewes' number e^e^e^79 an integer? ★★

Author(s):

Conjecture  

Skewes' number $ e^{e^{e^{79}}} $ is not an integer.

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Decomposing an even tournament in directed paths. ★★★

Author(s): Alspach; Mason; Pullman

Conjecture   Every tournament $ D $ on an even number of vertices can be decomposed into $ \sum_{v\in V}\max\{0,d^+(v)-d^-(v)\} $ directed paths.

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Direct product of reloids is a complete lattice homomorphism ★★

Author(s):

Conjecture  

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Yu Gi Oh Duel Links Cheats Generator 2024 Real Working (new method) ★★

Author(s):

Yu Gi Oh Duel Links Cheats Generator 2024 Real Working (new method)

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Dragon City Cheats Generator 2023-2024 Edition (Verified) ★★

Author(s):

Dragon City Cheats Generator 2023-2024 Edition (Verified)

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Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

Strict inequalities for products of filters

Author(s): Porton

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B}   \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A}   \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $. Particularly, is this formula true for $ \mathcal{A} = \mathcal{B} = \Delta \cap \uparrow^{\mathbb{R}} \left( 0 ; +   \infty \right) $?

A weaker conjecture:

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B}   \subset \mathcal{A} \ltimes \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $.

Keywords: filter products

Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW) ★★

Author(s):

Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW)

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Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused) ★★

Author(s):

Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused)

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Fishdom Cheats Generator Cheats Generator 2023-2024 (Free!!) ★★

Author(s):

Fishdom Cheats Generator Cheats Generator 2023-2024 (Free!!)

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Hungry Shark World Cheats Generator IOS Android No Verification 2024 (fresh method) ★★

Author(s):

Hungry Shark World Cheats Generator IOS Android No Verification 2024 (fresh method)

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Simpsons Tapped Out Cheats Generator Unlimited Cheats Generator IOS Android 2024 (get codes) ★★

Author(s):

Simpsons Tapped Out Cheats Generator Unlimited Cheats Generator IOS Android 2024 (get codes)

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Wall-Sun-Sun primes and Fibonacci divisibility ★★

Author(s):

Conjecture   For any prime $ p $, there exists a Fibonacci number divisible by $ p $ exactly once.

Equivalently:

Conjecture   For any prime $ p>5 $, $ p^2 $ does not divide $ F_{p-\left(\frac p5\right)} $ where $ \left(\frac mn\right) $ is the Legendre symbol.

Keywords: Fibonacci; prime

The Riemann Hypothesis ★★★★

Author(s): Riemann

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)

Keywords: Millenium Problems; zeta

Simpsons Tapped Out Cheats Generator (New Working Cheats Generator 2024) ★★

Author(s):

Simpsons Tapped Out Cheats Generator (New Working Cheats Generator 2024)

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Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.

Keywords:

Termination of the sixth Goodstein Sequence

Author(s): Graham

Question   How many steps does it take the sixth Goodstein sequence to terminate?

Keywords: Goodstein Sequence

Legal SimCity BuildIt Cheats Generator No Human Verification 2024 (No Surveys Needed) ★★

Author(s):

Legal SimCity BuildIt Cheats Generator No Human Verification 2024 (No Surveys Needed)

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"Working Cheats" Subway Surfers Coins Keys Generator Ios Android 2024 ★★

Author(s):

"Working Cheats" Subway Surfers Coins Keys Generator Ios Android 2024

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List colorings of edge-critical graphs ★★

Author(s): Mohar

Conjecture   Suppose that $ G $ is a $ \Delta $-edge-critical graph. Suppose that for each edge $ e $ of $ G $, there is a list $ L(e) $ of $ \Delta $ colors. Then $ G $ is $ L $-edge-colorable unless all lists are equal to each other.

Keywords: edge-coloring; list coloring

Covering systems with big moduli ★★

Author(s): Erdos; Selfridge

Problem   Does for every integer $ N $ exist a covering system with all moduli distinct and at least equal to~$ N $?

Keywords: covering system

World of Warships Cheats Generator Link 2024 (Cheats Generator that work) ★★

Author(s):

World of Warships Cheats Generator Link 2024 (Cheats Generator that work)

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Atomicity of the poset of completary multifuncoids ★★

Author(s): Porton

Conjecture   The poset of completary multifuncoids of the form $ (\mathscr{P}\mho)^n $ is for every sets $ \mho $ and $ n $:
    \item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

The sum of the two largest eigenvalues (Solved) ★★

Author(s):

The sum of the two largest eigenvalues (Solved)

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Rank vs. Genus ★★★

Author(s): Johnson

Question   Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?

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Polignac's Conjecture ★★★

Author(s): de Polignac

Conjecture   Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

In particular, this implies:

Conjecture   Twin Prime Conjecture: There are an infinite number of twin primes.

Keywords: prime; prime gap

Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends) ★★

Author(s):

Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends)

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The Hodge Conjecture ★★★★

Author(s): Hodge

Conjecture   Let $ X $ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $ X $.

Keywords: Hodge Theory; Millenium Problems

Lords Mobile Latest Cheats Version 2024 Free Gems Coins (WORKING GENERATOR) ★★

Author(s):

Lords Mobile Latest Cheats Version 2024 Free Gems Coins (WORKING GENERATOR)

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Family Island Cheats Generator 2024 Free No Verification (New.updated) ★★

Author(s):

Family Island Cheats Generator 2024 Free No Verification (New.updated)

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Continous analogue of Hirsch conjecture ★★

Author(s): Deza; Terlaky; Zinchenko

Conjecture   The order of the largest total curvature of the primal central path over all polytopes defined by $ n $ inequalities in dimension $ d $ is $ n $.

Keywords: curvature; polytope

Fundamental group torsion for subsets of Euclidean 3-space ★★

Author(s): Ancient/folklore

Problem   Does there exist a subset of $ \mathbb R^3 $ such that its fundamental group has an element of finite order?

Keywords: subsets of euclidean space; torsion

Edge-antipodal colorings of cubes ★★

Author(s): Norine

We let $ Q_d $ denote the $ d $-dimensional cube graph. A map $ \phi : E(Q_d) \rightarrow \{0,1\} $ is called edge-antipodal if $ \phi(e) \neq \phi(e') $ whenever $ e,e' $ are antipodal edges.

Conjecture   If $ d \ge 2 $ and $ \phi : E(Q_d) \rightarrow \{0,1\} $ is edge-antipodal, then there exist a pair of antipodal vertices $ v,v' \in V(Q_d) $ which are joined by a monochromatic path.

Keywords: antipodal; cube; edge-coloring

The Ultimate Guide to Gardenscapes Cheats and Hacks: Boost Your Game in 2024 ★★

Author(s):

Conjecture  

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Acyclic list colouring of planar graphs. ★★★

Author(s): Borodin; Fon-Der-Flasss; Kostochka; Raspaud; Sopena

Conjecture   Every planar graph is acyclically 5-choosable.

Keywords:

Point sets with no empty pentagon

Author(s): Wood

Problem   Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $ n $ vertices' trees, the star with $ n $ vertices has the most endomorphisms, while the path with $ n $ vertices has the least endomorphisms.

Keywords:

Ramsey properties of Cayley graphs ★★★

Author(s): Alon

Conjecture   There exists a fixed constant $ c $ so that every abelian group $ G $ has a subset $ S \subseteq G $ with $ -S = S $ so that the Cayley graph $ {\mathit Cayley}(G,S) $ has no clique or independent set of size $ > c \log |G| $.

Keywords: Cayley graph; Ramsey number