# 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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### Conjecture on Infinitely Many Consecutive Pairs of Early Primes

An early prime is one which is less than the arithmetic mean of the prime before and the prime after. Conjecture: There are infinitely many consecutive pairs of early primes MY attempt Well, the fact ...
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### Conjectures involving $\Lambda(n)$

As the title suggests, I am looking for conjectures involving the Von Mangoldt function, $\Lambda(n)$. I understand this is not a rigorous mathematical question, however if reference requests for ...
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### Lower bound of $n$ th Taxi-cab number $N = a^3 + b^3 = x^3 + y^3$

Let $N,a,b,x,y$ be distinct positive integers such that $$N = a^3 + b^3 = x^3 + y^3$$ Also known as Taxicab numbers or Taxi-cab numbers. see also : https://oeis.org/A001235 let $t(n)$ be the $n$ th ...
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### Conjectures about the greatest common divisor of a vertical column of the pascal triangle.

I was playing around with pascal triangle I noticed an interesting property concerning the greatest common divisor $gcd$ of binomial coefficients along a vertical line. Specifically, the line ...
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### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Now asked on MO here. Given the length of the sides of a quadrilateral $a,b,c,d$ the area of the quadrilateral is less than or equal to $\frac{(a+b+c+d)^2}{16}$ i.e it is an upper bound of the area ...
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### How do I just this conjecture?

I have 6 equations: $X_1=As+B_1t_1+B_2t_2$ $X_2=As+B_1t_2+B_2t_1$ $X_3=A_3s+B_3t_3$ $Y_1=C(X_1+t_1)$ $Y_2=C(X_2+t_2)$ $Y_3=D(X_3+t_3)$ where $X_1,X_2,X_3,Y_1,Y_2,Y_3,s,t_1,t_2,t_3$ are variables and ...
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### Consecutive numbers

Are there ever more consecutive composite numbers than there are primes up to that point? I imagine not, because the primes are the ones which cancel out multiples, so will inevitably have to fill in ...
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### Conjecture: Two different random triangles (both based on random points on a circle) have the same distribution of side length ratios.

On a circle, choose three uniformly random points $A,B,C$. Triangle $T_1$ has vertices $A,B,C$. The side lengths of $T_1$ are, in random order, $a,b,c$. Triangle $T_2$ is formed by drawing tangents to ...
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### How do you find constants in an k-Tuple conjectures?

By introducing modular objects associated to the sequel of rings $$(Z/2,Z/6,Z/30,Z/210,Z/2310,..., Z/p_n\#Z)$$ a sequence of coefficients is updated$$(2;\color{green}{\frac83}; 3.2;...$$ (see my ...
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### Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1)$

Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
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### Is $N=3\cdot 2^k+1$ prime if and only if $2^{N-1}\equiv 1 \pmod N$?

Is the following statement true? Let $k\geq 1$ be an integer and $N=3\cdot 2^k+1$. Then $N$ is a prime prime if and only if $2^{N-1}\equiv 1 \pmod {N}$ One implication is simply a Fermat's Little ...
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