[Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang | Hacker News](https://news.ycombinator.com/item?id=33512338)

[Some stuff I found interesting about number theory research | Hacker News](https://news.ycombinator.com/item?id=27955372)
[Ben Kuhn on X: "I had a fascinating conversation yesterday with a former roommate who's now a postdoc in number theory. I learned a lot about how math research works! Some stuff I found interesting:" / X](https://twitter.com/benskuhn/status/1419281153983500290)

[The Remarkable Number 1/89 (2004) | Hacker News](https://news.ycombinator.com/item?id=24929726)
[The remarkable number 1/89](http://www2.math.ou.edu/~dmccullough/teaching/miscellanea/miner.html)

[Back in 1993, I was taking a number theory class | Hacker News](https://news.ycombinator.com/item?id=27031242)
[Eric Lengyel on X: "Back in 1993, I was taking a number theory class, and there was a semester-long factorization contest that we could participate in. I implemented a distributed multiple polynomial quadratic sieve (MPQS) for the Mac, and I needed a cluster of computers to run it on." / X](https://twitter.com/EricLengyel/status/1389106103179378689)

[number theory summary | Britannica](https://www.britannica.com/summary/number-theory)

[Hacker's Guide to Numerical Analysis | Hacker News](https://news.ycombinator.com/item?id=26516081)
[A Universe of Sorts](https://web.archive.org/web/20230211055502/http://bollu.github.io/a-hackers-guide-to-numerical-analysis.html)

[Handbook of Graph Drawing and Visualization (2013) | Hacker News](https://news.ycombinator.com/item?id=29734063)
[Handbook of Graph Drawing and Visualization](https://cs.brown.edu/people/rtamassi/gdhandbook/)

[Hasse principle - Wikipedia](https://en.wikipedia.org/wiki/Hasse_principle)

[Number theory explained from first principles](https://explained-from-first-principles.com/number-theory/)

[The definitive guide to "modulo bias" and how to avoid it (2020) | Hacker News](https://news.ycombinator.com/item?id=32849145)
[The definitive guide to "Modulo Bias and how to avoid it"! - Kudelski Security Research](https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/)

## what math is

Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.

Mathematics is the extension of common sense by other means.

Without the rigorous structure that math provides, common sense can lead you astray.

Pythagorean's philosophy was a chunky stew of things we'd now call mathematics, things we'd now call religion, and things we'd now call mental illness.

If the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn't object.

George Berkeley denounced Newton's infinitesimals in a tone of high mockery sadly absent from current mathematical literature.

There are good choices and there are bad ones. In the mathematical context, the good choices are the ones that settle unnecessary perplexities without creating new ones.

What's required in many situations is not more facts - we're inundated already - but a better command of known facts.

## why people hate math

Most priesthoods (mathematicians included) are inclined to hide behind a wall of mystery and to commune only with their fellow priests.

People personalize events excessively, resisting an external perspective.
Since numbers and an impersonal view of the world are intimately related, this resistance contributes to an almost willful innumeracy.

They're often attracted to New Age beliefs, since these provide them with personally customized pronouncements.

The beauty of pure mathematics "cold and austere".
The sham romanticism inherent in the trite phrase "coldly rational" (as if "warmly rational" were some kind of oxymoron).

## combinatorial coefficients

The multiplication principle is deceptively simple and very important:
If some choice can be made in M different ways and some subsequent choice can be made in N different ways, then there are M × N different ways these choices can be made in succession.

The number of possible license plates in a state whose plates all have two letters followed by four numbers is 262 × 104.

People don't generally appreciate how large such seemingly tidy collections can be.

31 different flavors of ice cream? The number of possible triple-scoop cones without any repetition of flavors is therefore 31 × 30 × 29
If we're not interested in how the flavors are arranged on the cone but merely in how many three-flavored cones there are, we divide 26,970 by 6, to get 4,495 cones.
The reason we divide by 6 is that there are 6 = 3 × 2 × 1 different ways to arrange the three flavors

Lotteries choose six numbers out of a possible forty. (40 × 39 × 38 × 37 × 36 × 35)
If, however, we are interested not in the order then we divide by 720, since there are 720 (= 6 × 5 × 4 × 3 × 2 × 1) ways to arrange the six numbers

Same in all three examples:
(32 × 30 × 29)/(3 × 2 × 1) different three-flavored ice-cream cones
(40 × 39 × 38 × 37 × 36 × 35)/(6 × 5 × 4 × 3 × 2 × 1) different ways to choose six numbers out of forty
(52 × 51 × 50 × 49 × 48)/(5 × 4 × 3 × 2 × 1) different poker hands.

Numbers obtained in this way are called combinatorial coefficients.
They arise when we're interested in the number of ways of choosing R elements out of N elements and we're not interested in the order of the R elements chosen.

## history - proofs and mathematicians

0.33333…=1/3
Multiply both sides by 3 and you'll see 0.99999…=3/3=1
There's a whole field of mathematics that specializes in contemplating numbers of this kind, called nonstandard analysis.
.9 + .09 + .009 + .0009 + …
That pesky ellipsis is the real problem.
In the real world, you can never have infinitely many.
What's the numerical value of an infinite sum? It doesn't have one - until we give it one.
We should simply define the value of the infinite sum to be 1.

When you're field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something's wrong with your method.

In mathematics, you very seldom get the clearest account of an idea from the person who invented it.

[New Foundations is consistent - a difficult mathematical proof proved using Lean | Hacker News](https://news.ycombinator.com/item?id=40130924)
[New Foundations is consistent | Consistency of New Foundations](https://leanprover-community.github.io/con-nf//)

## math problems

demonstrate some math problems, such as the collatz conjecture

## numerical analysis

Estimate:
* how fast human hair grows in miles per hour
* approximately how many people die on earth each day
* how many cigarettes are smoked annually in this country
* what the population of the United States is
* the approximate distance from coast to coast
* what percentage of the world is Chinese
* How many pizzas are consumed each year in the United States?
* How many words have you spoken in your life?
* How many different people's names appear in The New York Times each year?
* How many watermelons would fit inside the U.S. Capitol building?
* How long would it take dump trucks to cart away Mount Fuji, to ground level? Assume trucks come every fifteen minutes, twenty-four hours a day, are instantaneously filled with mountain dirt and rock, and leave without getting in each other's way.

Practice estimating whatever quantity piques your curiosity.

Information can be gleaned from the barest numerical facts, and claims can often be refuted on the basis of these raw numbers alone.

To get a handle on big numbers, it's useful to come up with one or two collections corresponding to each power of ten, up to maybe 13 or 14.

Taking a human being to be spherical and about a meter in diameter:
* The size of a human cell is to that of a person as a person's size is to that of Rhode Island.
* A virus is to a person as a person is to the earth
* An atom is to a person as a person is to the earth's orbit around the sun
* A proton is to a person as a person is to the distance to Alpha Centauri.

Genesis says of the Flood that "… all the high hills that were under the whole heaven were covered…" Taken literally, this seems to indicate that there were 10,000 to 20,000 feet of water on the surface of the earth, equivalent to more than half a billion cubic miles of liquid! Since, according to biblical accounts, it rained for forty days and forty nights, or for only 960 hours, the rain must have fallen at a rate of at least fifteen feet per hour, certainly enough to sink any aircraft carrier.

## primes

Every positive number can be expressed in just one way as a product of prime numbers.
This is why we don't take 1 to be a prime, though some mathematicians have done so in the past; it breaks the uniqueness, because if 1 counts as prime, 60 could be written as 2 × 2 × 3 × 5 and 1 × 2 × 2 × 3 × 5 and 1 × 1 × 2 × 2 × 3 × 5…

The primes are the atoms of number theory, the basic indivisible entities of which all numbers are made.

Primes are not random, but it turns out that in many ways they act as if they were.

Pairs of primes that are separated by only 2, like 3-5 and 11-13, and called twin primes.

Goldbach conjecture: every even number greater than 2 is the sum of two primes.

## counting

[Computer scientists invent an efficient new way to count | Hacker News](https://news.ycombinator.com/item?id=40379175)
[Quanta Magazine](https://www.quantamagazine.org/computer-scientists-invent-an-efficient-new-way-to-count-20240516/)

## prime numbers

[113: Riemann-Zeta - explain xkcd](https://www.explainxkcd.com/wiki/index.php/113:_Riemann-Zeta)

[Graduate Student's Side Project Proves Prime Number Conjecture | Hacker News](https://news.ycombinator.com/item?id=31640297)
[Quanta Magazine](https://www.quantamagazine.org/graduate-students-side-project-proves-prime-number-conjecture-20220606/)

[Teenager solves stubborn riddle about prime number look-alikes | Hacker News](https://news.ycombinator.com/item?id=33192074)
[Quanta Magazine](https://www.quantamagazine.org/teenager-solves-stubborn-riddle-about-prime-number-look-alikes-20221013/)

[Mathematicians Find a New Class of Digitally Delicate Primes | Quanta Magazine](https://www.quantamagazine.org/mathematicians-find-a-new-class-of-digitally-delicate-primes-20210330)

[Breakthrough a step toward revealing hidden structure of prime numbers | Hacker News](https://news.ycombinator.com/item?id=41126944)
[‘Sensational breakthrough’ marks step toward revealing hidden structure of prime numbers | Science | AAAS](https://www.science.org/content/article/sensational-breakthrough-marks-step-toward-revealing-hidden-structure-prime-numbers)

## imaginary numbers

[410: Math Paper - explain xkcd](https://www.explainxkcd.com/wiki/index.php/410:_Math_Paper)

[217: e to the pi Minus pi - explain xkcd](https://www.explainxkcd.com/wiki/index.php/217:_e_to_the_pi_Minus_pi)

[179: e to the pi times i - explain xkcd](https://www.explainxkcd.com/wiki/index.php/179:_e_to_the_pi_times_i)

## grand unified programming theory

[GitHub - linpengcheng/PurefunctionPipelineDataflow: My Blog: The Math-based Grand Unified Programming Theory: The Pure Function Pipeline Data Flow with principle-based Warehouse/Workshop Model](https://github.com/linpengcheng/PurefunctionPipelineDataflow)
[GitHub - linpengcheng/PurefunctionPipelineDataflow: My Blog: The Math-based Grand Unified Programming Theory: The Pure Function Pipeline Data Flow with principle-based Warehouse/Workshop Model](https://github.com/linpengcheng/PurefunctionPipelineDataflow#Explainable-AI-System)

## combinatorics

[Application of combinatorics | Britannica](https://www.britannica.com/summary/combinatorics)

## theoretical computer science

[The zombie misconception of theoretical computer science | Hacker News](https://news.ycombinator.com/item?id=40912684)
[Shtetl-Optimized » Blog Archive » The Zombie Misconception of Theoretical Computer Science](https://scottaaronson.blog/?p=8106)

## optimization and applied maths

[optimization summary | Britannica](https://www.britannica.com/summary/optimization)

## numerical analysis

[numerical analysis summary | Britannica](https://www.britannica.com/summary/numerical-analysis)

[Concept of analysis in mathematics | Britannica](https://www.britannica.com/summary/analysis-mathematics)

## philosophy of math

[Conterintuitive facts in mathematics, CS, and physics | Hacker News](https://news.ycombinator.com/item?id=28764416)
[The most counterintuitive facts in all of mathematics, computer science, and physics](https://axisofordinary.substack.com/p/the-most-counterintuitive-facts-in)

[The Inconsistency of Arithmetic (2011) | Hacker News](https://news.ycombinator.com/item?id=28204040)
[The Inconsistency of Arithmetic | The n-Category Café](https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html)

[What do numbers look like? | Hacker News](https://news.ycombinator.com/item?id=33580136)
[web.archive.org/web/20231002085724/https://johnhw.github.io/umap_primes/index.md.html](https://web.archive.org/web/20231002085724/https://johnhw.github.io/umap_primes/index.md.html)

[What Gödel Discovered | Hacker News](https://news.ycombinator.com/item?id=25115746)
[What Gödel Discovered](https://stopa.io/post/269)

[You want chaos:Mathematics ends in contradiction:6 proofs : ConfrontingChaos](https://old.reddit.com/r/ConfrontingChaos/comments/yb2xa0/you_want_chaosmathematics_ends_in_contradiction6/)

## set theory

[set theory summary | Britannica](https://www.britannica.com/summary/set-theory)

[Google researcher, long out of math, cracks devilish problem about sets | Hacker News](https://news.ycombinator.com/item?id=34236889)
[Quanta Magazine](https://www.quantamagazine.org/long-out-of-math-an-ai-programmer-cracks-a-pure-math-problem-20230103/)

['A-team' of math proves a critical link between addition and sets | Hacker News](https://news.ycombinator.com/item?id=38545522)
[Quanta Magazine](https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/)

## real analysis

[ClassicalRealAnalysis](http://classicalrealanalysis.info/)