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Are there infinitely many Mersenne primes? by

Ciro Santilli 40 Updated 2025-07-16

Admissible prime k-tuple by

Ciro Santilli 40 Updated 2025-07-16

Prime k-tuple conjecture by

Ciro Santilli 40 Updated 2025-07-16

There are infinitely many prime k-tuples for every admissible tuple.

Generalization of the Twin prime conjecture.

As of 2023, there was no specific admissible tuple for which it had been proven that there infinite of, only bounds of type:

there are infinitely 2-tuple instances with at most a finite bound

But these do not specify which specific tuple, e.g. Yitang Zhang's theorem.

Twin prime conjecture by

Ciro Santilli 40 Updated 2025-07-16

Let's show them how it's done with primes + awk. Edit. They have a -d option which also shows gaps!!! Too strong:

sudo apt install bsdgames
primes -d 1 100 | awk '/\(2\)/{print $1 - 2, $1 }'

gives us the list of all twin primes up to 100:

Tested on Ubuntu 22.10.

Sums of three cubes by

Ciro Santilli 40 Updated 2025-07-16

Prime number theorem by

Ciro Santilli 40 Updated 2025-07-16

prime-number-theorem by

Ciro Santilli 40 Updated 2025-07-16

Consider this is a study in failed computational number theory.

The

n / l n (n)

approximation converges really slowly, and we can't easy go far enough to see that the ration converges to 1 with only awk and primes:

sudo apt intsall bsdgames
cd prime-number-theorem
./main.py 100000000

Runs in 30 minutes tested on Ubuntu 22.10 and P51, producing:

Figure 1.
Linear $π (n)$ vs $n / l n (n)$ approximation plot
. $f (n) = n$ and $f (n) = l o g (n)$ are added to give a better sense of scale. $l o g (n)$ is too close to 0 and not visible, and the approximation almost overlaps entirely with $p i$ .

Figure 2.
$π (n) - n / l n (n)$
. It is clear that the difference diverges, albeit very slowly.

Figure 3.
$π (n) / (n / l n (n))$
. We just don't have enough points to clearly see that it is converging to 1.0, the convergence truly is very slow. The logarithm integral approximation is much much better, but we can't calculate it in awk, sadface.

But looking at: en.wikipedia.org/wiki/File:Prime_number_theorem_ratio_convergence.svg we see that it takes way longer to get closer to 1, even at

1 0^{24}

it is still not super close. Inspecting the code there we see:

(* Supplement with larger known PrimePi values that are too large for \
Mathematica to compute *)
LargePiPrime = {{10^13, 346065536839}, {10^14, 3204941750802}, {10^15,
     29844570422669}, {10^16, 279238341033925}, {10^17,
    2623557157654233}, {10^18, 24739954287740860}, {10^19,
    234057667276344607}, {10^20, 2220819602560918840}, {10^21,
    21127269486018731928}, {10^22, 201467286689315906290}, {10^23,
    1925320391606803968923}, {10^24, 18435599767349200867866}};

so OK, it is not something doable on a personal computer just like that.

Direct3D by

Ciro Santilli 40 Updated 2025-07-16

Best Illusion of the Year Contest by

The Best Illusion of the Year Contest is an annual competition that showcases and celebrates innovative optical illusions created by artists, researchers, and enthusiasts from around the world. Established in 2005, the contest is typically organized by the Neural Correlate Society and aims to highlight the intersection of art and science, particularly in the realm of visual perception. Participants submit their original illusions, which can involve a variety of techniques and concepts to manipulate how we perceive visual information.

Gravitational mirage by

A gravitational mirage, more commonly referred to as gravitational lensing, is a phenomenon that occurs when the gravitational field of a massive object, such as a galaxy or cluster of galaxies, distorts the light from objects that are located behind it. This effect arises from the predictions of Einstein's General Theory of Relativity, which posits that mass can curve spacetime. When light from a distant star or galaxy passes near a massive object, the object's gravity bends the light's path.

Operations research awards by

Operations Research (OR) awards are recognitions given to individuals or teams for significant contributions to the field of operations research, which involves the application of advanced analytical methods to help make better decisions. These awards can recognize various aspects of the field, including research papers, innovative methodologies, applications of OR techniques in real-world problems, and contributions to the education and practice of operations research.

Semiotics of fashion by

Semiotics of fashion refers to the study of signs and symbols within the context of fashion, exploring how clothing, accessories, and other fashion elements communicate meaning. This field examines how different elements of fashion function as signs that convey information, emotions, identities, and cultural values.

Government Operational Research Service by

The Government Operational Research Service (GORS) is a part of the UK government that employs operational research (OR) principles to inform and enhance decision-making within public sector organizations. GORS is focused on applying analytical and quantitative methods to complex problems in areas such as policy development, resource allocation, and service delivery.

Trace (semiology) by

In semiology, or semiotics, "trace" refers to the concept of a mark or sign that signifies something else. The term is often associated with the work of semiotic theorists like Ferdinand de Saussure, who laid the groundwork for modern semiotics by discussing how signs operate within a system of meaning. In semiotics, a sign consists of the signifier (the form that the sign takes) and the signified (the concept it represents).

Visual semiotics by

Visual semiotics is the study of signs and symbols in visual forms of communication, focusing on how images, symbols, and visual elements convey meaning. It draws from semiotics, the broader theory of signs and meaning-making, established by theorists like Ferdinand de Saussure and Charles Sanders Peirce. In visual semiotics, the emphasis is on analyzing how visual representations—such as photographs, paintings, advertisements, films, and other visual media—function as a system of signs.

Clitic climbing by

Clitic climbing is a linguistic phenomenon that occurs in certain languages where a clitic, which is a type of morpheme that behaves like a word but cannot stand alone (such as contractions or auxiliary verbs in English), can "climb" over another element in a sentence to attach to a different position, typically a higher one, than its original position.

Extraposition by

Extraposition is a linguistic phenomenon in which a clause or phrase that would typically appear in a more typical syntactic position is instead placed later in the sentence, often for the purpose of emphasizing other elements or improving the flow of the sentence. In English, extraposition is especially common with certain types of clauses, such as that-clauses or wh-clauses. For example, instead of saying: 1. "That the meeting was canceled surprised everyone.

Lexicalist hypothesis by

The Lexicalist Hypothesis is a theoretical concept in linguistics that posits that many grammatical functions and structures are primarily determined by the properties and organization of the lexicon—the mental repository of words and their meanings—rather than by the syntactic structures themselves. Essentially, this hypothesis argues that much of syntax is directly driven by the lexical entries of words, which contain not only the word's meaning but also information about its grammatical behavior, such as its syntactic category (e.g.

Musical syntax by

Musical syntax refers to the rules and structures that govern how musical elements are organized, similar to how grammatical rules govern the structure of sentences in language. It encompasses the relationships between musical notes, chords, rhythms, and phrases, helping to create coherence and meaning within a piece of music. Key aspects of musical syntax include: 1. **Melody and Harmony**: The organization of pitches (melodic lines) and how these interact with chords (harmony) to create tension and resolution.