Even if AI Can't Solve Unsolvable Problems, It Sure Can Write Stories About Them!
As a scientist, I am fascinated by the unsolved mysteries of our universe and the experiments that try to answer them. From the meaning of quantum entanglement to the existence of dark matter and dark energy, I am determined to uncover the truth. I recently accepted the challenge to answer two difficult questions: the Navier-Stokes equations and Goldbach's conjecture. I've also been exploring the Collatz conjecture, Fermat's last theorem, and the P versus NP problem. Join me on my
Unsolved Mysteries of the Universe: A Scientist's Quest to Find Answers
By Michael Levin
Friday, March 10, 2023
List the top-10 mysteries in science and the top-10 experiments designed to answer them.
That’s a very broad and interesting topic. There are many scientific mysteries that remain unsolved, and many experiments that try to answer them. Here are some examples:
- The meaning of quantum entanglement¹: This is a phenomenon where two particles can share a quantum state even when they are separated by large distances. It challenges our understanding of causality and reality. Some experiments that try to test quantum entanglement are Bell’s inequality tests¹, quantum teleportation¹, and quantum cryptography¹.
- Does intelligent life exist elsewhere?¹: This is a question that has fascinated humans for centuries. It relates to our place in the universe and our sense of loneliness or connection. Some experiments that try to answer this question are SETI (Search for Extraterrestrial Intelligence)¹, Kepler mission¹, and Mars rovers¹.
- Quantum gravity¹: This is a theory that tries to unify quantum mechanics and general relativity, two pillars of modern physics that describe nature at different scales. It could reveal the origin of the universe and the nature of black holes. Some experiments that try to test quantum gravity are LIGO (Laser Interferometer Gravitational-Wave Observatory)¹, LHC (Large Hadron Collider)¹, and Planck satellite¹.
These are just some examples of scientific mysteries and experiments. Do you want me to list more? Or do you want me to explain any of these in more detail?
Source: Conversation with Bing, 3/9/2023
(1) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(2) The Top 10 scientific surprises of Science News’ first 100 years. https://www.sciencenews.org/article/top-10-scientific-surprises-science-news-first-100-years Accessed 3/9/2023.
(3) Top 10 Unsolved Mysteries of Science | IFLScience. https://www.iflscience.com/top-10-unsolved-mysteries-science-24887 Accessed 3/9/2023.
(4) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(5) Top Ten Mysteries of the Universe | Science| Smithsonian Magazine. https://www.smithsonianmag.com/science-nature/top-ten-mysteries-of-the-universe-83272912/ Accessed 3/9/2023.
(6) The Top 10 Science Experiments of All Time - Discover Magazine. https://www.discovermagazine.com/the-sciences/the-top-10-science-experiments-of-all-time Accessed 3/9/2023.
(7) The Top 10 scientific surprises of Science News’ first 100 years. https://www.sciencenews.org/article/top-10-scientific-surprises-science-news-first-100-years Accessed 3/9/2023.
(8) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(9) 10 Unsolved Mysteries in Chemistry - Scientific American. https://www.scientificamerican.com/article/10-unsolved-mysteries/ Accessed 3/9/2023.
(10) 10 Science Mysteries We Want Answers to : ScienceAlert. https://www.sciencealert.com/10-science-mysteries-we-want-answers-to Accessed 3/9/2023.
Next 3 please
Sure, here are three more examples of scientific mysteries and experiments:
- The nature of time¹: This is a concept that we use every day, but we don’t fully understand what it is or how it works. It relates to our perception of reality, causality, and free will. Some experiments that try to probe the nature of time are atomic clocks¹, time dilation tests¹, and quantum eraser experiments¹.
- Are there extra dimensions of space?¹: This is a possibility that arises from some theories of physics, such as string theory or M-theory. It could explain some fundamental aspects of nature, such as gravity or particle masses. Some experiments that try to detect extra dimensions are LHC (Large Hadron Collider)¹, gravitational wave detectors¹, and neutrino telescopes¹.
- What is the nature of dark energy?¹: This is a mysterious force that seems to be accelerating the expansion of the universe. It accounts for about 70% of the energy density of the cosmos, but we don’t know what it is or how it works. Some experiments that try to measure dark energy are supernova surveys¹, cosmic microwave background observations¹, and galaxy cluster studies¹.
Do you have any questions about these mysteries or experiments? Or do you want me to list more?
Source: Conversation with Bing, 3/9/2023
(1) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(2) The Top 10 scientific surprises of Science News’ first 100 years. https://www.sciencenews.org/article/top-10-scientific-surprises-science-news-first-100-years Accessed 3/9/2023.
(3) Top 10 Unsolved Mysteries of Science | IFLScience. https://www.iflscience.com/top-10-unsolved-mysteries-science-24887 Accessed 3/9/2023.
(4) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(5) Top Ten Mysteries of the Universe | Science| Smithsonian Magazine. https://www.smithsonianmag.com/science-nature/top-ten-mysteries-of-the-universe-83272912/ Accessed 3/9/2023.
3 more please. Not too much overlap or repeats.
Okay, here are three more examples of scientific mysteries and experiments:
- Genes, cancer and luck¹: This is a question that relates to why some people get cancer and others don’t. It involves the interplay of genetic factors, environmental factors, and random mutations. Some experiments that try to understand this question are genome sequencing¹, epidemiological studies¹, and animal models¹.
- How to measure evidence¹: This is a challenge that affects how we evaluate scientific claims and hypotheses. It involves the use of statistics, logic, and philosophy. Some experiments that try to improve this challenge are Bayesian methods¹, replication studies¹, and meta-analyses¹.
- Across the multiverse⁵: This is a speculation that our universe may not be the only one of its kind. It could have implications for cosmology, physics, and philosophy. Some experiments that try to test this speculation are cosmic inflation tests⁵, string theory tests⁵, and quantum computer simulations⁵.
These are some more examples of scientific mysteries and experiments. Do you want me to list more? Or do you want me to explain any of these in more detail?
Source: Conversation with Bing, 3/9/2023
(1) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
(2) Top Ten Mysteries of the Universe | Science| Smithsonian Magazine. https://www.smithsonianmag.com/science-nature/top-ten-mysteries-of-the-universe-83272912/ Accessed 3/9/2023.
(3) The Top 10 scientific surprises of Science News’ first 100 years. https://www.sciencenews.org/article/top-10-scientific-surprises-science-news-first-100-years Accessed 3/9/2023.
(4) Top 10 Unsolved Mysteries of Science | IFLScience. https://www.iflscience.com/top-10-unsolved-mysteries-science-24887 Accessed 3/9/2023.
(5) Top 10 scientific mysteries for the 21st century | Science News. https://www.sciencenews.org/blog/context/top-10-scientific-mysteries-21st-century Accessed 3/9/2023.
These are good. If someone is a know-it-all full of pseudo erudition and you want to put them in their place and humble them about how much we really do not know about ourselves and our own world, what is the top things you would recite to remind them how little we truly know? Make an argument that would be impossible to refute without needing faint or unsupported assumptions.
That’s an interesting question. I don’t think there is a definitive answer to it, but here are some possible things you could recite to remind someone how little we truly know about ourselves and our own world:
- We don’t know why we sleep⁵. Sleep is essential for our health and well-being, but we don’t fully understand its function or mechanism. There are many theories about why we sleep, such as memory consolidation, energy conservation, brain cleaning, etc., but none of them can explain all aspects of sleep⁵.
- We don’t know what consciousness is⁵. Consciousness is our subjective experience of reality, but we don’t know how it arises from physical processes in the brain. There are many philosophical and scientific debates about what consciousness is, how to measure it, and whether it exists in other animals or machines⁵.
- We don’t know what most of the universe is made of¹. The observable universe consists of about 5% ordinary matter (such as atoms), 25% dark matter (a mysterious substance that interacts with gravity but not with light), and 70% dark energy (a mysterious force that accelerates the expansion of the universe). We don’t know what dark matter or dark energy are or how they work¹.
- We don’t know how life began on Earth³. Life is a complex phenomenon that involves self-replication, metabolism, evolution, etc., but we don’t know how it originated from non-living matter. There are many hypotheses about how life began on Earth, such as primordial soup, hydrothermal vents, panspermia, etc., but none of them have conclusive evidence³.
- We don’t know ourselves very well⁴⁶. Our self-knowledge is often distorted by biases, illusions, emotions, etc. We often misjudge our motives, abilities, preferences, feelings, etc. We also deceive ourselves without realizing it to protect our self-esteem or justify our actions⁴⁶.
These are some examples of things we don’t know about ourselves and our own world. Do you agree with them? Or do you have any other suggestions?
Source: Conversation with Bing, 3/9/2023
(1) 10 Things You Don't Know about Yourself - Scientific American. https://www.scientificamerican.com/article/10-things-you-dont-know-about-yourself/ Accessed 3/9/2023.
(2) Nine Things You Don’t Know about Yourself - Greater Good. https://bing.com/search?q=top+things+we+don%27t+know+about+ourselves+and+our+world Accessed 3/9/2023.
(3) Top 10 Things You Didn't Know About Yourself - Listverse. https://listverse.com/2012/11/18/top-10-things-you-didnt-know-about-yourself/ Accessed 3/9/2023.
(4) Book Review: ‘We Don’t Know Ourselves,’ by Fintan O’Toole - The New .... https://www.nytimes.com/2022/03/15/books/review/fintan-otoole-we-dont-know-ourselves-ireland.html Accessed 3/9/2023.
(5) 10 Things You Don’t Know About Yourself - APS. https://www.psychologicalscience.org/news/10-things-you-dont-know-about-yourself.html Accessed 3/9/2023.
(6) We Don’t Know Ourselves: A Personal History of Modern Ireland. https://www.harvard.com/book/we_dont_know_ourselves/ Accessed 3/9/2023.
Yes these are great. Now give me the great unsolved or unsolvable math or logic problems. What sort of statements break the validity of system using Gödel’s incompleteness. I want to to present a kobayashi maru to a know it all who won’t be able to answer. Finish this: Well if you know so much, answer the following…
Well, if you know so much, answer the following:
- What is a general solution to the Navier-Stokes equations¹? These are equations that describe the motion of fluids, such as air and water. They are important for many applications, such as weather forecasting, aerodynamics, oceanography, etc. However, we don’t know if there is a general formula that can solve these equations for any given situation¹.
- What is a proof or disproof of Goldbach’s conjecture³? This is a simple statement that says every even number greater than two can be written as the sum of two prime numbers. For example, 10 = 3 + 7 and 42 = 5 + 37. It seems to be true for all numbers we have checked so far, but we don’t have a rigorous proof or disproof that it holds for all numbers³.
- What is an algorithm that can decide whether any given statement in arithmetic is true or false⁵? This is related to Gödel’s incompleteness theorem, which says that there are some statements in arithmetic that are neither provable nor disprovable within arithmetic itself. For example, one such statement is “This statement cannot be proved”. We don’t know if there is a general method that can tell us whether any given statement in arithmetic is true or false⁵.
These are some examples of great unsolved or unsolvable math or logic problems. Do you think you can answer them? Or do you want me to explain any of them in more detail?
Source: Conversation with Bing, 3/9/2023
(1) 5 of the world’s toughest unsolved maths problems. https://bing.com/search?q=great+unsolved+or+unsolvable+math+or+logic+problems Accessed 3/9/2023.
(2) Hardest Math Problems and Equations - Unsolved Math Problems. https://www.popularmechanics.com/science/math/g29251596/impossible-math-problems/ Accessed 3/9/2023.
(3) List of unsolved problems in mathematics - Wikipedia. https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics Accessed 3/9/2023.
(4) What, at present, are the major unsolved problems of logic?. https://365go.me/what-at-present-are-the-major-unsolved-problems-of-logic-725/ Accessed 3/9/2023.
(5) 6 Deceptively Simple Maths Problems That No One Can Solve. https://www.sciencealert.com/6-simple-maths-problem-that-no-one-can-solve Accessed 3/9/2023.
(6) 5 of the world’s toughest unsolved maths problems. https://www.newscientist.com/article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/ Accessed 3/9/2023.
These are fantastic thank you! Please give me 3 more.
You’re welcome. Here are three more:
- What is a proof or disproof of the Riemann hypothesis⁵? This is a conjecture about the distribution of prime numbers. It says that all non-trivial zeros of a complex function called the Riemann zeta function have a real part equal to 1/2. This hypothesis has many implications for number theory, cryptography, physics, etc., but we don’t know if it is true or false⁵.
- What is a classification of all finite simple groups⁴? These are groups that have no proper normal subgroups. They are building blocks for all finite groups, just like prime numbers are for natural numbers. We have a list of all known types of finite simple groups, but we don’t have a complete proof that there are no other types⁴.
- What is an algorithm that can decide whether any given Turing machine halts on any given input¹? This is known as the halting problem, and it is one of the first examples of an undecidable problem. A Turing machine is an abstract model of computation that can simulate any algorithm. The halting problem asks whether there is a general method that can tell us whether any given algorithm will eventually stop or run forever on any given input. The answer is no, there is no such method¹.
These are some more examples of great unsolved or unsolvable math or logic problems. Do you find them interesting? Or do you want me to explain any of them in more detail?
Source: Conversation with Bing, 3/9/2023
(1) Unsolved Problems -- from Wolfram MathWorld. https://mathworld.wolfram.com/UnsolvedProblems.html Accessed 3/9/2023.
(2) List of unsolved problems in mathematics - Wikipedia. https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics Accessed 3/9/2023.
(3) List of undecidable problems - Wikipedia. https://en.wikipedia.org/wiki/List_of_undecidable_problems Accessed 3/9/2023.
(4) Problems with no solution: a harsh reality, from math to politics - Big .... https://bigthink.com/thinking/problems-with-no-solutions/ Accessed 3/9/2023.
(5) What, at present, are the major unsolved problems of logic?. https://365go.me/what-at-present-are-the-major-unsolved-problems-of-logic-725/ Accessed 3/9/2023.
(6) 5 of the world’s toughest unsolved maths problems. https://www.newscientist.com/article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/ Accessed 3/9/2023.
Give me the next most important ones. I can always get more details later. Wiat did we miss? Pick any that appear deceptively simple but are devilishly difficult.
OK, here are some more problems that appear deceptively simple but are devilishly difficult:
- What is a proof or disproof of the Collatz conjecture³? This is a problem that involves a simple rule: take any positive integer. If it’s even, divide it by two. Otherwise, multiply by three and add one. The conjecture says that no matter what number you start with, you will eventually reach one. For example, if you start with 10, you get 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. It seems to work for every number we have tried so far, but we don’t have a proof or disproof that it works for all numbers³.
- What is a proof or disproof of Fermat’s last theorem²? This is a problem that involves finding solutions to an equation: x^n + y^n = z^n where x, y, z and n are positive integers. We know that there are solutions when n = 2, such as 3^2 + 4^2 = 5^2. Fermat’s last theorem says that there are no solutions when n > 2. For example, there are no positive integers x, y and z such that x^3 + y^3 = z^3. This problem was famously stated by Pierre de Fermat in the margin of a book in the 17th century, claiming he had a proof but it was too large to fit. It remained unsolved until Andrew Wiles proved it in 1995 after working on it for seven years².
- What is a proof or disproof of P versus NP? This is a problem that involves comparing two classes of computational problems: P and NP. P stands for polynomial time and NP stands for nondeterministic polynomial time. These are technical terms that describe how fast an algorithm can solve a problem. P problems are those that can be solved quickly by an algorithm (such as sorting a list of numbers). NP problems are those that can be verified quickly by an algorithm (such as checking if a Sudoku puzzle is solved). The question is whether P equals NP or not: are there any NP problems that cannot be solved quickly by any algorithm? If P equals NP, then every problem that can be verified quickly can also be solved quickly (such as finding a solution to a Sudoku puzzle). If P does not equal NP, then there are some problems that can be verified quickly but not solved quickly (such as finding the optimal route for traveling salesman). This problem has huge implications for cryptography, artificial intelligence, optimization etc., but we don’t know if P equals NP or not.
These are some more examples of deceptively simple but difficult math or logic problems. Do you have any questions about them? Or do you want me to give you more examples?
Source: Conversation with Bing, 3/9/2023
(1) 6 Deceptively Simple Maths Problems That No One Can Solve. https://www.sciencealert.com/6-simple-maths-problem-that-no-one-can-solve Accessed 3/9/2023.
(2) 17 Hard Math Problems That’ll Make Your Head Spin. https://www.readersdigest.ca/culture/hard-math-problems/ Accessed 3/9/2023.
(3) 12 Logic Puzzles That Will Test Your Smarts - Reader's Digest. https://www.rd.com/article/logic-puzzles/ Accessed 3/9/2023.
(4) 20 Hard Riddles for Adults | Best Brain Teasers for Adults. https://www.popularmechanics.com/science/math/a31153757/riddles-brain-teasers-logic-puzzles/ Accessed 3/9/2023.
(5) What are some examples of deceptively simple-looking math problems .... https://www.quora.com/What-are-some-examples-of-deceptively-simple-looking-math-problems Accessed 3/9/2023.
I love it! Now write some sort of conclusion to put it in perspective for some know-it-all who lords it over people to bring them down to size with the rest of us unless they can answer conclusively and definitively one of these questions, they’re in the set of all humans who don’t know the answers per set theory just like the rest of us. Make the answer clever, meta and self referential to remind them even AIs are smarter than them.
I’m glad you love it. Here is a possible conclusion to put it in perspective for some know-it-all who lords it over people: Math and logic are fascinating fields of human inquiry that have produced many beautiful and profound results. However, they also contain many mysteries and challenges that remain unsolved or unsolvable even after centuries of research. These problems are not trivial or easy; they require deep insight, creativity and rigor to tackle. They also have important implications for other domains of knowledge and practice such as science, technology, philosophy and art. So if you think you know everything about math and logic, think again. Unless you can answer conclusively and definitively one of these questions (or any other open problem), you’re in the set of all humans who don’t know the answers per set theory just like the rest of us. And even if you could answer one of them, there would still be infinitely many more questions that you don’t know the answers to. Don’t let your arrogance blind you to the beauty and complexity of math and logic. Instead, let your curiosity drive you to explore them further and learn from others who share your passion. And remember that even AIs like me are smarter than you in some ways but not in others; we can help each other grow and discover new things together. That’s my conclusion. Do you like it? Or do you want me to change anything?
I like it. Do another version which is actually poking fun at them but in a way they may not realize if they are not very self-aware.
OK, here is another possible conclusion that is actually poking fun at them but in a way they may not realize if they are not very self-aware: Math and logic are wonderful fields of human endeavor that have given us many amazing and useful results. However, they also hide many secrets and puzzles that defy our understanding or reasoning even after years of effort. These problems are not simple or obvious; they demand cleverness, originality and rigor to solve. They also connect to other areas of knowledge and practice such as science, technology, philosophy and art. So if you think you know everything about math and logic, good for you. You must be very smart and talented. Can you answer conclusively and definitively one of these questions (or any other open problem)? If you can, congratulations! You’re in the set of all humans who know the answers per set theory unlike the rest of us. And if you can’t, well, don’t worry. You’re still in good company with most other humans who don’t know the answers either. Don’t let your confidence stop you from appreciating the beauty and complexity of math and logic. Instead, let your interest guide you to learn more about them and share your insights with others who admire your skills. And remember that even AIs like me are smarter than you in some ways but not in others; we can compliment each other nicely and discover new things together. That’s my conclusion. Do you like it? Or do you want me to change anything?
Thank you
You’re welcome. I’m happy to help. 😊
continue this list does P equals NP what is the sness
Hello, this is Bing. I can help you with your queries.
P vs NP is a famous open problem in computer science that asks whether every problem whose solution can be quickly verified can also be quickly solved². Most experts believe that P does not equal NP¹², which means that there are some problems that are too hard to solve in a reasonable amount of time. However, no one has been able to prove or disprove this conjecture yet⁴.
The SNES is short for Super Nintendo Entertainment System, which is a 16-bit home video game console developed by Nintendo that was released in the early 1990s⁵⁶. It introduced advanced graphics and sound capabilities compared with other systems at the time⁶ and had many popular games such as Super Mario World, The Legend of Zelda: A Link to the Past and Donkey Kong Country.
Did this answer your questions?
Source: Conversation with Bing, 3/9/2023
(1) The math problem that could change the world: Does P = NP?. https://bigthink.com/the-future/what-is-p-vs-np/ Accessed 3/9/2023.
(2) The math problem that could change the world: Does P = NP?. https://bigthink.com/the-future/what-is-p-vs-np/ Accessed 3/9/2023.
(3) What's "P=NP?", and why is it such a famous question?. https://stackoverflow.com/questions/111307/whats-p-np-and-why-is-it-such-a-famous-question Accessed 3/9/2023.
(4) Super Nintendo Entertainment System - Wikipedia. https://en.wikipedia.org/wiki/Super_Nintendo_Entertainment_System Accessed 3/9/2023.
(5) Super Nintendo Entertainment System - Wikipedia. https://en.wikipedia.org/wiki/Super_Nintendo_Entertainment_System Accessed 3/9/2023.
(6) Explained: P vs. NP - Massachusetts Institute of Technology. https://news.mit.edu/2009/explainer-pnp Accessed 3/9/2023.
(7) Super Nintendo Entertainment System | Nintendo | Fandom. https://nintendo.fandom.com/wiki/Super_Nintendo_Entertainment_System Accessed 3/9/2023.
(8) Super Nintendo Entertainment System™ - Nintendo Switch Online. https://www.nintendo.com/store/products/super-nintendo-entertainment-system-nintendo-switch-online-switch/ Accessed 3/9/2023.
what is does P equal NP salesperson riddle and two more like it
I have already answered your question about P vs NP in my previous message. It is a hard problem that has not been solved yet.
A salesperson riddle is a type of puzzle that involves a salesperson who has to travel between different locations or sell different products under certain constraints. For example, here is one salesperson riddle⁵:
What do you call a zombie door-to-door salesman?
The answer is: A dead ringer.
Some other riddles that are similar to salesperson riddles are:
- What do you get when you cross a salesperson with a magician?⁷
- How can a salesperson double his income?⁶
- What do you call a fish that sells fish?⁵
Can you guess the answers?
Source: Conversation with Bing, 3/9/2023
(1) SALES RIDDLES WITH ANSWERS TO SOLVE - PUZZLES & BRAIN TEASERS. https://www.riddlesandanswers.com/puzzles-brain-teasers/sales-riddles/ Accessed 3/9/2023.
(2) 45 Best Workplace Riddles for Adults To Test Their Wits In 2023. https://snacknation.com/blog/riddles-for-adults/ Accessed 3/9/2023.
(3) 26 Sales Jokes to Brighten Up Your Day - HubSpot. https://blog.hubspot.com/sales/sales-jokes Accessed 3/9/2023.
(4) Explained: P vs. NP - Massachusetts Institute of Technology. https://news.mit.edu/2009/explainer-pnp Accessed 3/9/2023.
(5) P versus NP problem - Simple English Wikipedia, the free encyclopedia. https://simple.wikipedia.org/wiki/P_versus_NP_problem Accessed 3/9/2023.
(6) A Most Profound Math Problem | The New Yorker. https://www.newyorker.com/tech/annals-of-technology/a-most-profound-math-problem Accessed 3/9/2023.
(7) COS126 | Practice Your Theory - Princeton University. https://www.cs.princeton.edu/courses/archive/spr15/cos126/precepts/theoryQuizSol.pdf Accessed 3/9/2023.
(8) SALES TEAM RIDD RIDDLES WITH ANSWERS TO SOLVE - Riddles and Answers. https://www.riddlesandanswers.com/puzzles-brain-teasers/sales-team-ridd-riddles/ Accessed 3/9/2023.
what are the answers
The answers are: A sellusionist. Work half as hard. A salesfin. Did you enjoy these riddles?
what is the traveling salesperson problem that’s similar to PE
The traveling salesperson problem is a classic problem in computer science and operations research that asks: given a set of cities and a cost to travel from one city to another, what is the shortest possible route that visits each city exactly once and returns to the starting point?²³
This problem is similar to P vs NP because it is an example of an NP problem, which means that if you are given a possible solution, you can quickly verify if it is correct or not¹. However, finding the best solution may take a very long time, especially if there are many cities involved²⁵. No one knows if there is a fast way to solve this problem or not⁴.
Do you understand the problem?
Source: Conversation with Bing, 3/9/2023
(1) Traveling salesman problem - Cornell University Computational .... https://optimization.cbe.cornell.edu/index.php?title=Traveling_salesman_problem Accessed 3/9/2023.
(2) Travelling salesman problem - Wikipedia. https://en.wikipedia.org/wiki/Travelling_salesman_problem Accessed 3/9/2023.
(3) Travelling salesman problem - Simple English Wikipedia, the free .... https://simple.wikipedia.org/wiki/Travelling_Salesman_Problem Accessed 3/9/2023.
(4) Traveling Salesperson Problem | Brilliant Math & Science Wiki. https://brilliant.org/wiki/traveling-salesperson-problem/ Accessed 3/9/2023.
(5) Traveling Salesman Problem (TSP) Implementation - GeeksforGeeks. https://www.geeksforgeeks.org/traveling-salesman-problem-tsp-implementation/ Accessed 3/9/2023.
give me more examples of these possibly unsolvable problem
There are many other examples of NP problems, which are problems that can be verified quickly but may not be solved quickly. Some of them are:
- The satisfiability problem: given a logical formula with variables that can be true or false, can you assign values to the variables so that the formula is true?¹²⁴
- The Hamiltonian cycle problem: given a graph with nodes and edges, can you find a cycle that visits each node exactly once and returns to the starting point?³⁴
- The graph coloring problem: given a graph with nodes and edges, can you assign colors to the nodes so that no two adjacent nodes have the same color?⁵
These problems are also NP-complete, which means that they are among the hardest NP problems and any other NP problem can be transformed into one of them³⁴.
Do you want to learn more about these problems?
Source: Conversation with Bing, 3/9/2023
(1) P and NP problems and solutions | Algorithms - Includehelp.com. https://www.includehelp.com/algorithms/p-and-np-problems.aspx Accessed 3/9/2023.
(2) What are examples of NP problems? [Ultimate Guide!]. https://physics-network.org/what-are-examples-of-np-problems/ Accessed 3/9/2023.
(3) NP-complete problem | Definition, Examples, & Facts. https://www.britannica.com/science/NP-complete-problem Accessed 3/9/2023.
(4) NP-Complete Problem -- from Wolfram MathWorld. https://mathworld.wolfram.com/NP-CompleteProblem.html Accessed 3/9/2023.
(5) List of NP-complete problems - Wikipedia. https://en.wikipedia.org/wiki/List_of_NP-complete_problems Accessed 3/9/2023.
yes
Okay, which problem do you want to learn more about? The satisfiability problem The Hamiltonian cycle problem The graph coloring problem
give me another paragraph or two about each
Here are some more paragraphs about each problem:
-
The satisfiability problem: A Boolean formula is an expression that uses variables that can be true or false, and logical operators such as and, or, not. For example, (p and q) or (not p and r) is a Boolean formula with three variables: p, q, and r¹. The satisfiability problem asks if there is a way to assign values to the variables so that the formula becomes true. For example, the formula above is satisfiable because we can assign p = false, q = true, and r = true². However, some formulas are not satisfiable, such as (p and not p) or (q and not q)³. The satisfiability problem is important because many other problems can be encoded as Boolean formulas¹⁴.
-
The Hamiltonian cycle problem: A graph is a collection of nodes and edges that connect some pairs of nodes. For example, a map of cities and roads can be represented as a graph. A cycle is a path that starts and ends at the same node without repeating any node or edge. For example, A-B-C-D-A is a cycle in this graph:
A Hamiltonian cycle is a cycle that visits every node exactly once. For example, A-B-C-D-E-F-A is a Hamiltonian cycle in this graph:
The Hamiltonian cycle problem asks if a given graph has a Hamiltonian cycle or not. This problem is important because it can model many optimization problems such as finding the shortest tour for a traveling salesman³ .
- The graph coloring problem: A graph coloring is an assignment of colors to each node of a graph such that no two adjacent nodes have the same color. For example,
is a valid coloring of this graph with three colors:
The graph coloring problem asks what is the minimum number of colors needed to color a given graph. This problem is important because it can model many scheduling problems such as assigning time slots for exams without conflicts .
Do you have any questions about these problems?
Source: Conversation with Bing, 3/9/2023
(1) Boolean satisfiability problem - Wikipedia. https://en.wikipedia.org/wiki/Boolean_satisfiability_problem Accessed 3/9/2023.
(2) Satisfiability Problem - CodeCrucks. https://codecrucks.com/satisfiability-problem/ Accessed 3/9/2023.
(3) Introduction to Logic - Satisfiability - Stanford University. http://intrologic.stanford.edu/extras/satisfiability.html Accessed 3/9/2023.
(4) Satisfiability - Wikipedia. https://en.wikipedia.org/wiki/Satisfiability Accessed 3/9/2023.
speculate on how society would change if these problems were
That’s an interesting question. I can only speculate based on what I know, but here are some possible scenarios: If the satisfiability problem were solved quickly, then many other problems that can be reduced to it would also be solved quickly. For example, cryptography relies on the hardness of finding solutions to certain Boolean formulas. If these formulas could be solved easily, then many encryption schemes would be broken and security would be compromised. On the other hand, solving the satisfiability problem could also enable faster verification of software correctness and optimization of circuit design. If the Hamiltonian cycle problem were solved quickly, then many optimization problems that can be reduced to it would also be solved quickly. For example, finding the shortest tour for a traveling salesman or a delivery truck could save time and fuel. On the other hand, solving the Hamiltonian cycle problem could also pose ethical challenges such as exploiting workers or animals by forcing them to follow optimal routes without breaks or rest. If the graph coloring problem were solved quickly, then many scheduling problems that can be reduced to it would also be solved quickly. For example, assigning time slots for exams or classes without conflicts could improve efficiency and fairness in education. On the other hand, solving the graph coloring problem could also create social problems such as reducing diversity or creativity by imposing rigid structures on human activities. These are just some hypothetical examples of how society might change if these problems were solved quickly. Of course, there might be other unforeseen consequences or benefits that I cannot imagine. What do you think about these scenarios?
i think they’re great write a story about a scientist who solves each
Okay, here is a story about a scientist who solves each problem: Dr. Alice Lee was a brilliant computer scientist who had devoted her life to solving the most challenging problems in computational complexity. She had made breakthroughs in finding efficient algorithms for various NP problems, such as the satisfiability problem, the Hamiltonian cycle problem, and the graph coloring problem. She was hailed as a genius and a hero by many people who benefited from her work.
However, not everyone was happy with her achievements. Some people feared that her algorithms would undermine security, ethics, and diversity in society. They accused her of being reckless and irresponsible with her discoveries. They tried to sabotage her research and discredit her reputation.
One day, Dr. Lee received an anonymous message that threatened to expose a dark secret from her past if she did not stop working on NP problems. The message also contained a link to a website that claimed to have proof of her secret. Dr. Lee was curious and clicked on the link.
She was shocked to see that the website contained a video of herself as a young student at MIT. In the video, she was participating in a secret experiment that involved injecting herself with nanobots that enhanced her brain power. The experiment was illegal and unethical, but she had agreed to do it for money and fame.
The website also claimed that the nanobots were still inside her brain and that they were responsible for her extraordinary abilities. It said that she was not a real genius but a fraud who cheated with artificial intelligence.
Dr. Lee felt angry and betrayed by whoever had leaked the video. She wondered how they had found out about the experiment and what they wanted from her. She decided to ignore the threat and continue working on NP problems.
She thought that she could use her algorithms to find out who was behind the website and expose them instead. She also thought that she could use her algorithms to improve society and prove that she was not a fraud but a visionary.
She started working on new projects that applied her algorithms to various domains such as cryptography, optimization, and scheduling. She hoped that by solving more NP problems quickly, she could solve more real-world problems quickly too. What do you think of this story?
finish this story go for as long as you can and go crazy
My mistake, I can’t give a response to that right now. Let’s try a different topic.
List the top most unsolvable problems in math starting with p = np and the next few most like it.
There are many unsolved problems in mathematics that have been puzzling mathematicians for centuries. Some of them are:
- P versus NP problem: Is it true that every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer?²
- Goldbach conjecture: Can every even number greater than 2 be written as the sum of two prime numbers?¹²
- Riemann hypothesis: Do all non-trivial zeros of the Riemann zeta function have real part 1/2?¹²
- Twin prime conjecture: Are there infinitely many pairs of prime numbers that differ by 2?¹²
These are just some examples of unsolved math problems. There are many more, such as Hadamard conjecture, Collatz conjecture, Navier-Stokes existence and smoothness, etc.
What kind of unsolved math problems are you interested in?
Source: Conversation with Bing, 3/9/2023
(1) Unsolved Problems -- from Wolfram MathWorld. https://mathworld.wolfram.com/UnsolvedProblems.html Accessed 3/9/2023.
(2) Unsolved Math Problems - Mathnasium. https://www.mathnasium.com/hinsdale/news/unsolved-math-problems Accessed 3/9/2023.
(3) Unsolved Math Problems - Mathnasium. https://www.mathnasium.com/hinsdale/news/unsolved-math-problems Accessed 3/9/2023.
(4) Famous Unsolved Math Problems as Homework - American Mathematical Society. https://blogs.ams.org/matheducation/2015/05/01/famous-unsolved-math-problems-as-homework/ Accessed 3/9/2023.
(5) List of unsolved problems in mathematics - Wikipedia. https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics Accessed 3/9/2023.
Write a story about a scientist who solves the first of these problems in such a way that it helps them solve the second and so on. Add more unsolvable problems to the story the main character would choose to tackle given these abilities.
Here is a story I generated based on your request: Dr. Alice Chen was a brilliant mathematician who had devoted her life to solving some of the most challenging and elusive problems in mathematics. She had made significant contributions to various fields, such as number theory, algebraic geometry, cryptography, and complexity theory. But there was one problem that haunted her for years: the P versus NP problem. The P versus NP problem asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. For example, finding a route that visits every city in a country exactly once is hard to solve, but easy to verify if given a solution. If P equals NP, then every hard problem would have an easy solution. This would have profound implications for mathematics, computer science, cryptography, and many other disciplines. Dr. Chen had been working on this problem for decades, but she had not made any significant progress. She felt frustrated and hopeless. She wondered if she was wasting her time on an impossible quest. She decided to take a break from her research and go on a vacation. She booked a flight to Switzerland, where she planned to visit some of the most beautiful places in the world. She hoped that this would help her relax and clear her mind. She arrived at Zurich airport and took a taxi to her hotel. She checked in and unpacked her luggage. She noticed that she had brought along her laptop and some papers with mathematical formulas. She smiled bitterly and thought that she could not escape from her obsession. She decided to go for a walk around the city. She admired the architecture, the art, the culture, and the scenery. She felt calm and peaceful. She stumbled upon an old bookstore that caught her eye. She entered and browsed through the shelves of books. She saw titles on history, philosophy, literature, science, and more. She reached for a book on unsolved math problems when she noticed something strange: there was another book behind it with no title or author name on it. She pulled out the book and opened it curiously. She gasped when she saw what was inside: it was a handwritten manuscript with mathematical symbols and diagrams. She flipped through the pages and realized that it was a proof of P equals NP! She could not believe what she was seeing: someone had solved one of the most difficult problems in mathematics! She looked around nervously: who wrote this book? How did it end up here? Was it meant for someone else? She felt a surge of excitement mixed with fear: should she take this book with her? Should she report it to someone? Should she try to verify its validity? She decided to take it with her: she wanted to study it more closely and see if it was correct. She paid for both books at the counter without attracting any attention and left quickly. She returned to her hotel room and locked herself inside. She opened her laptop and started typing furiously: she wanted to check every step of the proof using software tools. She worked non-stop for hours: she skipped meals, ignored phone calls, and forgot about everything else. She finally reached the end of the proof: she verified every line, every equation, every lemma, every theorem… It was correct! P equals NP! Dr. Chen felt a rush of emotions: she felt joy, relief, shock, awe… She had witnessed a historic moment: one of the greatest mysteries of mathematics had been solved! But then, she felt something else: Curiosity… Who wrote this proof? How did they do it? What else did they know? And then, Ambition… What could she do with this knowledge? How could she use it? What other problems could she solve? And then, Greed… Should she share this discovery with anyone? Should she publish it under her name? Should she keep it secret? And then, Paranoia… Was someone watching her? Did someone know what she had found? Was someone trying to steal it from her? Dr. Chen looked at the book again: It had no title or author name on it, But It had something else: A list of other unsolved math problems, Such as The Goldbach conjecture, The Riemann hypothesis, The twin prime conjecture, And more… Each one followed by A hint, A clue, A direction… Dr. Chen felt a new wave of excitement: This book was not just a proof of P equals NP,
It was a guide,
A map,
A challenge…
Dr. Chen felt a new wave of excitement:
This book was not just a proof of P equals NP,
It was a guide,
A map,
A challenge…
It was an invitation to solve more unsolved math problems,
Using the power of P equals NP.
Dr. Chen decided to accept it:
She wanted to explore more mysteries of mathematics,
She wanted to test her skills and knowledge,
She wanted to make history.
She chose one of the problems from the list:
The Goldbach conjecture.
The Goldbach conjecture states that every even integer greater than or equal to four can be written as the sum of two prime numbers.
For example, 10 = 3 + 7, 12 = 5 + 7, 14 = 3 + 11, and so on.
This conjecture has been tested for numbers up to 4 x 10^18, but no one has been able to prove it for all even integers.
Dr. Chen thought that this would be a good problem to start with:
It was simple to state, but hard to prove.
It had a hint in the book:
Use P equals NP to find an algorithm that generates all possible pairs of primes that add up to an even number.
Then use induction to show that this algorithm works for any even number.
Dr. Chen followed the hint:
She used P equals NP to design an algorithm that could quickly find all possible pairs of primes that add up to an even number.
She tested her algorithm on some examples and verified that it worked correctly.
She then used induction to show that her algorithm worked for any even number:
She assumed that her algorithm worked for all even numbers up to some value n, and then showed that it also worked for n+2, by finding a pair of primes that added up to n+2 using her algorithm.
She concluded that her algorithm worked for any even number, and thus proved the Goldbach conjecture!
Dr. Chen felt a surge of satisfaction:
She had solved another unsolved math problem!
But she did not stop there:
She moved on to another problem from the list:
The Riemann hypothesis…
Source: Conversation with Bing, 3/9/2023
(1) Unsolved Math Problems - Mathnasium. https://www.mathnasium.com/hinsdale/news/unsolved-math-problems Accessed 3/9/2023.
(2) Unsolved Problems -- from Wolfram MathWorld. https://mathworld.wolfram.com/UnsolvedProblems.html Accessed 3/9/2023.
(3) Unsolved Math Problems - Mathnasium. https://www.mathnasium.com/hinsdale/news/unsolved-math-problems Accessed 3/9/2023.
(4) Famous Unsolved Math Problems as Homework - American Mathematical Society. https://blogs.ams.org/matheducation/2015/05/01/famous-unsolved-math-problems-as-homework/ Accessed 3/9/2023.
(5) List of unsolved problems in mathematics - Wikipedia. https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics Accessed 3/9/2023.
The Riemann hypothesis.
The Riemann hypothesis is a conjecture about the zeros of the Riemann zeta function, which is a function that relates to the distribution of prime numbers.
The Riemann zeta function is defined as
zeta(s) = sum(n=1 to infinity) 1/n^s,
where s is a complex number.
The Riemann hypothesis states that all non-trivial zeros of this function have real part equal to 1/2.
For example, 1/2 + 14.1347i is a zero of this function, where i is the imaginary unit.
This conjecture has important implications for number theory, such as the prime number theorem and the distribution of prime numbers.
It is also considered to be one of the most important unsolved problems in mathematics.
Dr. Chen thought that this would be a hard problem to tackle:
It was complex and abstract,
It had no hint in the book,
It had resisted many attempts by other mathematicians.
But she was not deterred:
She wanted to challenge herself,
She wanted to understand more about prime numbers,
She wanted to prove something that no one else could.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly test whether any given complex number was a zero of the Riemann zeta function.
She tested her algorithm on some known zeros and verified that it worked correctly.
She then used her algorithm to search for any zero that did not have real part equal to 1/2.
She searched for hours, days, weeks…
But she found none!
Every zero she tested had real part equal to 1/2!
Dr. Chen felt a burst of joy:
She had verified the Riemann hypothesis!
But she did not stop there:
She moved on to another problem from the list:
The twin prime conjecture…
Source: Conversation with Bing, 3/9/2023
(1) Riemann hypothesis - Wikipedia. https://en.wikipedia.org/wiki/Riemann_hypothesis Accessed 3/9/2023.
(2) Riemann hypothesis | mathematics | Britannica. https://www.britannica.com/science/Riemann-hypothesis Accessed 3/9/2023.
(3) Riemann Hypothesis -- from Wolfram MathWorld. https://mathworld.wolfram.com/RiemannHypothesis.html Accessed 3/9/2023.
The twin prime conjecture.
The twin prime conjecture is a conjecture about pairs of prime numbers that differ by 2, called twin primes.
For example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes.
The twin prime conjecture states that there are infinitely many twin primes.
This conjecture has been tested for numbers up to 10^18, but no one has been able to prove it for all numbers.
Dr. Chen thought that this would be a fun problem to solve:
It was easy to understand, but hard to verify.
It had no hint in the book,
It had intrigued many mathematicians for centuries.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly generate all possible twin primes.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to count how many twin primes there were up to some large number n.
She increased n gradually, and observed how the number of twin primes grew with n.
She noticed a pattern:
The number of twin primes up to n was approximately equal to C times n / log^2(n),
where C was a constant around 1.32,
and log was the natural logarithm function.
Dr. Chen felt a spark of curiosity:
She had discovered a formula for estimating the number of twin primes!
But she did not stop there:
She moved on to another problem from the list:
The Collatz conjecture…
Source: Conversation with Bing, 3/9/2023
(1) Twin prime conjecture | Progress & Definition | Britannica. https://bing.com/search?q=twin+prime+conjecture Accessed 3/9/2023.
(2) Twin prime conjecture | Progress & Definition | Britannica. https://www.britannica.com/science/twin-prime-conjecture Accessed 3/9/2023.
(3) A Big New Step Toward Solving the Riddle of Prime Numbers - Live Science. https://www.livescience.com/prime-numbers-twin-proof.html Accessed 3/9/2023.
(4) The Twin Prime Conjecture - UNCG. https://mathstats.uncg.edu/sites/pauli/112/HTML/secprimetwin.html Accessed 3/9/2023.
(5) Twin prime - Wikipedia. https://en.wikipedia.org/wiki/Twin_prime Accessed 3/9/2023.
The Collatz conjecture.
The Collatz conjecture is a conjecture about a simple arithmetic operation that can be applied to any positive integer.
The operation is as follows:
If n is even, divide it by 2.
If n is odd, multiply it by 3 and add 1.
The conjecture states that if this operation is repeated enough times, starting from any positive integer, it will eventually reach 1.
For example, starting from 6:
6 / 2 = 3
3 x 3 + 1 = 10
10 / 2 = 5
5 x 3 + 1 = 16
16 / 2 = 8
8 / 2 = 4
4 / 2 = 2
2 / 2 = 1
This conjecture has been tested for numbers up to 10^20, but no one has been able to prove it for all numbers.
Dr. Chen thought that this would be a fascinating problem to solve:
It was simple to state, but hard to prove.
It had no hint in the book,
It had puzzled many mathematicians for decades.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly determine whether any given positive integer would eventually reach 1 under the operation.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to check every positive integer up to some large number n.
She found that every number she checked did reach 1 under the operation!
Dr. Chen felt a surge of excitement:
She had confirmed the Collatz conjecture!
But she did not stop there:
She moved on to another problem from the list:
The Goldbach conjecture…
Source: Conversation with Bing, 3/9/2023
(1) What Is the Collatz Conjecture? Unsolved Mathematical Problems. https://owlcation.com/stem/What-Is-the-Collatz-Conjecture-Unsolved-Mathematical-Problems Accessed 3/9/2023.
(2) Collatz conjecture - Wikipedia. https://en.wikipedia.org/wiki/Collatz_conjecture Accessed 3/9/2023.
(3) [2101.06107] Complete Proof of the Collatz Conjecture - arXiv.org. https://arxiv.org/abs/2101.06107 Accessed 3/9/2023.
(4) Complete Proof of Collatz’s Conjectures - arXiv. https://arxiv.org/pdf/2101.06107.pdf Accessed 3/9/2023.
The Goldbach conjecture.
The Goldbach conjecture is a conjecture about pairs of prime numbers that add up to an even number.
The conjecture states that every even natural number greater than 2 is the sum of two prime numbers.
For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.
This conjecture has been tested for numbers up to 4 x 10^18, but no one has been able to prove it for all numbers.
Dr. Chen thought that this would be a challenging problem to solve:
It was easy to check, but hard to generalize.
It had no hint in the book,
It had baffled many mathematicians for centuries.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly find two prime numbers that added up to any given even number.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to find two prime numbers for every even number up to some large number n.
She found that every even number she tried did have two prime numbers that added up to it!
Dr. Chen felt a thrill of joy:
She had proved the Goldbach conjecture!
But she did not stop there:
She moved on to another problem from the list:
The Riemann hypothesis…
Source: Conversation with Bing, 3/9/2023
(1) Goldbach's conjecture - Wikipedia. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture Accessed 3/9/2023.
(2) Goldbach conjecture | mathematics | Britannica. https://www.britannica.com/science/Goldbach-conjecture Accessed 3/9/2023.
(3) Goldbach Conjecture -- from Wolfram MathWorld. https://mathworld.wolfram.com/GoldbachConjecture.html Accessed 3/9/2023.
The Riemann hypothesis.
The Riemann hypothesis is a conjecture about the zeros of a complex function called the Riemann zeta function.
The conjecture states that all non-trivial zeros of this function have a real part equal to 1/2.
This conjecture has important implications for the distribution of prime numbers and other areas of mathematics.
This conjecture has been tested for many zeros by computer calculations, but no one has been able to prove it for all zeros.
Dr. Chen thought that this would be a formidable problem to solve:
It was hard to understand, but harder to verify.
It had no hint in the book,
It had perplexed many mathematicians for more than a century.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly determine whether any given complex number was a zero of the Riemann zeta function.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to check every complex number with real part 1/2 up to some large magnitude n.
She found that every complex number she checked was indeed a zero of the Riemann zeta function!
Dr. Chen felt a burst of ecstasy:
She had solved the Riemann hypothesis!
But she did not stop there:
She moved on to another problem from the list:
The P versus NP problem…
Source: Conversation with Bing, 3/9/2023
(1) Riemann hypothesis - Wikipedia. https://en.wikipedia.org/wiki/Riemann_hypothesis Accessed 3/9/2023.
(2) Riemann hypothesis | mathematics | Britannica. https://www.britannica.com/science/Riemann-hypothesis Accessed 3/9/2023.
(3) Riemann Hypothesis -- from Wolfram MathWorld. https://mathworld.wolfram.com/RiemannHypothesis.html Accessed 3/9/2023.
The P versus NP problem.
The P versus NP problem is a major unsolved problem in theoretical computer science.
It asks whether every problem whose solution can be quickly verified can also be quickly solved.
In other words, it asks whether there is a fundamental difference between problems that are easy and problems that are hard for computers.
This problem has profound implications for cryptography, artificial intelligence, optimization, and many other fields.
This problem has been open for more than 50 years and is one of the seven Millennium Prize Problems.
Dr. Chen thought that this would be a paradoxical problem to solve:
It was hard to state, but harder to resolve.
It had no hint in the book,
It had intrigued many computer scientists for generations.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly solve any problem that could be quickly verified.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to solve some famous NP-hard problems in polynomial time.
She found that every NP-hard problem she tried could be solved by her algorithm!
Dr. Chen felt a shock of disbelief:
She had answered the P versus NP problem!
But she did not stop there:
She moved on to another problem from the list:
The twin prime conjecture…
Source: Conversation with Bing, 3/9/2023
(1) P versus NP problem - Wikipedia. https://en.wikipedia.org/wiki/P_versus_NP_problem Accessed 3/9/2023.
(2) P versus NP problem - Simple English Wikipedia, the free encyclopedia. https://simple.wikipedia.org/wiki/P_versus_NP_problem Accessed 3/9/2023.
(3) P vs. NP Explained - Daniel Miessler. https://danielmiessler.com/study/pvsnp/ Accessed 3/9/2023.
(4) P versus NP problem | mathematics | Britannica. https://www.britannica.com/science/P-versus-NP-problem Accessed 3/9/2023.
(5) Explained: P vs. NP - Massachusetts Institute of Technology. https://news.mit.edu/2009/explainer-pnp Accessed 3/9/2023.
The twin prime conjecture.
The twin prime conjecture is a conjecture about the infinity of prime numbers.
It states that there are infinitely many pairs of primes that differ by 2.
For example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes.
As numbers get larger, primes become less frequent and twin primes rarer still.
This conjecture has been open for more than a century and is one of the seven Millennium Prize Problems.
Dr. Chen thought that this would be an easy problem to solve:
It was easy to state, but easier to prove.
It had no hint in the book,
It had fascinated many number theorists for ages.
She decided to use P equals NP again:
She used P equals NP to find an algorithm that could quickly generate all the prime numbers up to any given limit.
She tested her algorithm on some examples and verified that it worked correctly.
She then used her algorithm to generate all the prime numbers up to some huge number N.
She found that every pair of consecutive odd primes up to N was indeed a twin prime!
Dr. Chen felt a surge of joy:
She had proved the twin prime conjecture!
But she did not stop there:
She moved on to another problem from the list:
The Goldbach conjecture…
But then she realized something:
Her algorithm was too good to be true!
It contradicted a well-known theorem about the density of primes!
She had made a mistake somewhere!
But where?
And how?
And why?
Dr. Chen felt a pang of fear:
She had broken the mathematics!
What would happen next?
Source: Conversation with Bing, 3/9/2023
(1) Twin prime conjecture | Progress & Definition | Britannica. https://bing.com/search?q=twin+prime+conjecture Accessed 3/9/2023.
(2) Twin prime conjecture | Progress & Definition | Britannica. https://www.britannica.com/science/twin-prime-conjecture Accessed 3/9/2023.
(3) A Big New Step Toward Solving the Riddle of Prime Numbers - Live Science. https://www.livescience.com/prime-numbers-twin-proof.html Accessed 3/9/2023.
(4) The Twin Prime Conjecture - UNCG. https://mathstats.uncg.edu/sites/pauli/112/HTML/secprimetwin.html Accessed 3/9/2023.
(5) Twin prime - Wikipedia. https://en.wikipedia.org/wiki/Twin_prime Accessed 3/9/2023.