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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A function asymptotical equivalent with the prime counting function?

Let $p_n$ be the $n$-th prime number and $Q_a(N)$ be the number of primes of the form $p_n^2+a$ where $1\leq n\leq N$ and $a$ is positive and even. For some $a$ like $26,56$ it seems that no solutions ...
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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 ...
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Can two circles intersect each other at right angles, such that one circle passes through the center of the other circle?

More precisely, do there exist two intersecting circles such that the tangent lines at the intersection points make an angle of 90 degrees, and one of the circles passes through the center of the ...
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Is $3$ the only prime that is both a Mersenne prime and a Fermat prime?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part III

Preamble: This post is an offshoot of this earlier MSE question. The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by \$\...
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