## 🔖 Holyhedron | Wikipedia

Bookmarked Holyhedron (Wikipedia)

In mathematics, a holyhedron is a type of 3-dimensional geometric body: a polyhedron each of whose faces contains at least one polygon-shaped hole, and whose holes' boundaries share no point with each other or the face's boundary.

The concept was first introduced by John H. Conway; the term "holyhedron" was coined by David W. Wilson in 1997 as a pun involving polyhedra and holes. Conway also offered a prize of 10,000 USD, divided by the number of faces, for finding an example, asking:

Is there a polyhedron in Euclidean three-dimensional space that has only finitely many plane faces, each of which is a closed connected subset of the appropriate plane whose relative interior in that plane is multiply connected?

No actual holyhedron was constructed until 1999, when Jade P. Vinson presented an example of a holyhedron with a total of 78,585,627 faces; another example was subsequently given by Don Hatch, who presented a holyhedron with 492 faces in 2003, worth about 20.33 USD prize money.

## 🔖 Gilbreath’s conjecture | Wikipedia

Bookmarked Gilbreath's conjecture (Wikipedia)
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.

## 🔖 Look-and-say sequence | Wikipedia

Bookmarked Look-and-say sequence (Wikipedia)
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (sequence A005150 in the OEIS).
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11.
11 is read off as "two 1s" or 21.
21 is read off as "one 2, then one 1" or 1211.
1211 is read off as "one 1, one 2, then two 1s" or 111221.
111221 is read off as "three 1s, two 2s, then one 1" or 312211. The look-and-say sequence was introduced and analyzed by John Conway.

## 🔖 Surreal number | Wikipedia

Bookmarked Surreal numbers (Wikipedia)
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.[a] If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.

## 🔖 Sierpinski number | Wikipedia

Bookmarked Sierpiński number (Wikipedia)
In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that {\displaystyle k\times 2^{n}+1} is composite, for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all members of the following set are composite:
{\displaystyle \left\{\,k\cdot {}2^{n}+1:n\in \mathbb {N} \,\right\}.} ## 🔖 Sylvester’s Line Problem | Wolfram MathWorld

Read Sylvester's Line Problem (Wolfram MathWorld)

Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."

Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft et al. 1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald (1944). Coxeter (1948, 1969) transformed the problem into an elementary form, and a very short proof using the notion of Euclidean distance was given by Kelly (Coxeter 1948, 1969; Chvátal 2004). The theorem also follows using projective duality from a result of Melchior (1940) proved by a simple application of Euler's polyhedral formula (Chvátal 2004).

Additional information on the theorem can be found in Borwein and Moser (1990), Erdős and Purdy (1991), Pach and Agarwal (1995), and Chvátal (2003).

In September 2003, X. Chen proved a conjecture of Chvátal that, with a certain definition of a line, the Sylvester-Gallai theorem extends to arbitrary finite metric spaces.

## 🔖 Sylvester’s Problem, Steinberg’s Solution | Cut the Knot

Bookmarked Sylvester's Problem, Gallai's Solution (cut-the-knot.org)
T. Gallai's proof has been outlined by P. Erdös in his submission of the problem to The American Mathematical Monthly in 1943. Solution Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary.

## 🔖 Sylvester’s Problem, Steinberg’s Solution | Cut the Knot

Bookmarked Sylvester's Problem, Steinberg's Solution (cut-the-knot.org)
R. Steinberg's was actually the first published solution to Syvester's problem, Solution Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary. We consider the configuration in the projective plane.

## 🔖 Sylvester–Gallai theorem | Wikipedia

Bookmarked Sylvester–Gallai theorem (Wikipedia)

The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
* all the points lie on a single line; or
* there is a line which contains exactly two of the points.
It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.

A line that contains exactly two of a set of points is known as an ordinary line. According to a strengthening of the theorem, every finite point set (not all on a line) has at least a linear number of ordinary lines. There is an algorithm that finds an ordinary line in a set of n points in time proportional to n log n in the worst case.

## 🔖 The Erdős Discrepancy Problem (6.09.2017) | Terence Tao | YouTube

Bookmarked The Erdős Discrepancy Problem (6.09.2017) at Instytut Matematyczny Uniwersytetu Wrocławskiego by Terence Tao (YouTube)

The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n and d range over the natural numbers. In the 1930s, Erdős posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwiłł, as well as a surprising application of the Shannon entropy inequalities, the Erdős discrepancy problem was solved in 2015. In his talk TT will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.

## 🔖 Tao’s resolution of the Erdős discrepancy problem | AMS | K. Soundararajan

Bookmarked Tao’s resolution of the Erdős discrepancy problem by K. Soundararajan (Bulletin of the American Mathematical Society, Volume 55, Number 1, January 2018, Pages 81–92)

This article gives a simplified account of some of the ideas behind Tao’s resolution of the Erdős discrepancy problem.
http://dx.doi.org/10.1090/bull/1598 | PDF

## 🔖 The Entropy Decrement Method and the Erdos Discrepancy Problem | Simons Institute for the Theory of Computing

Bookmarked The Entropy Decrement Method and the Erdos Discrepancy Problem (Simons Institute for the Theory of Computing)

Tuesday, April 11th, 2017 9:30 am – 10:30 am
Structure vs. Randomness
Speaker: Terry Tao, UCLA

We discuss a variant of the density and energy increment arguments that we call an "entropy decrement method", which can be used to locate a scale in which two relevant random variables share very little mutual information, and thus behave somewhat like independent random variables.  We were able to use this method to obtain a new correlation estimate for multiplicative functions, which in turn was used to establish the Erdos discrepancy conjecture that any sequence taking values in {-1,+1} had unbounded sums on homogeneous arithmetic progressions.