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 Math Unsolved Problems:     more detail

1. Unsolved Problems -- From MathWorld
Klee, V. Some Unsolved Problems in Plane Geometry. Math. Mag. London Oliver Boyd,1966. Ogilvy, C. S. Tomorrow's math unsolved problems for the Amateur.
http://mathworld.wolfram.com/UnsolvedProblems.html
 Foundations of Mathematics Mathematical Problems Unsolved Problems Unsolved Problems There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture 2. The Riemann hypothesis 3. The 4. The conjecture that there exists a Hadamard matrix for every positive multiple of 4. 5. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes 6. Determination of whether NP-problems are actually P-problems 7. The Collatz problem 8. Proof that the 196-algorithm does not terminate when applied to the number 196. 9. Proof that 10 is a solitary number 10. Finding a formula for the probability that two elements chosen at random generate the symmetric group 11. Solving the happy end problem for arbitrary n 12. Finding an Euler brick whose space diagonal is also an integer. 13. Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers n such that , where is the totient function 15. Determining if the

2. Problem -- From MathWorld
Klee, V. Some Unsolved Problems in Plane Geometry. Math. Mag. ed. New York Dover,1954. Ogilvy, C. S. Tomorrow's math unsolved problems for the Amateur.
http://mathworld.wolfram.com/Problem.html
 Foundations of Mathematics Mathematical Problems Problem Collections History and Terminology ... Terminology Problem A problem is an exercise whose solution is desired. Mathematical "problems" may therefore range from simple puzzles to examination and contest problems to propositions whose proofs require insightful analysis. There are many unsolved problems in mathematics. Two famous problems which have recently been solved include Fermat's last theorem (by Andrew Wiles) and the Kepler conjecture (by T. C.Hales). Among the most prominent of remaining unsolved problems are the Goldbach conjecture Riemann hypothesis , the conjecture that there are an infinite number of twin primes , as well as many more. K.S. Brown, D. Eppstein, S. Finch, and C. Kimberling maintain extensive pages of unsolved problems in mathematics. Unsolved Problems References Artino, R. A.; Gaglione, A. M.; and Shell, N. The Contest Problem Book IV: Annual High School Mathematics Examinations 1973-1982. Washington, DC: Math. Assoc. Amer., 1982. Alexanderson, G. L.; Klosinski, L.; and Larson, L.

3. Unsolved Problems: References
New York 1966. Ogilvy 1972 C. Stanley Ogilvy, Tomorrow's math unsolved problemsfor the Amateur. 2nd edition. Oxford University Press. New York 1972.
http://cage.rug.ac.be/~hvernaev/problems/references.html
##### General References
The following books contain unsolved problems or
are referenced by the unsolved problem of the week
Especially rich are [Croft 1991] [Guy 1994] and [Klee 1991]
[Beiler 1966]
Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertain. 2nd edition. Dover. New York: 1966.
[Bondy 1976]
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications. North Holland. New York: 1976.
[Boroczky 1987]
Intuitive Geometry. North-Holland Publishing Company. New York: 1987.
[Croft 1991]
Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry. Springer-Verlag. New York: 1991.
[Dudeney 1970]
H. E. Dudeney, Amusements in Mathematics. Dover. New York: 1970.
[Dunham 1990]
William Dunham, Journey Through Genius: The Great Theorems of Mathematics. John Wiley and Sons. New York: 1990.
[Erdos 1980]
Old and New Problems and Results in Combinatorial Number Theory.
[Gardner 1978]
Martin Gardner, Mathematical Magic Show. Vintage Books. New York: 1978.
[Gardner 1983]
Martin Gardner

4. Unsolved Problems
Several people have asked me about unsolved problems. I will take the easy way out see the list Hill, NC 275993250 (rpratt@math.unc.edu). Julio Subocz notes that
http://www.math.fau.edu/locke/unsolved.htm
##### Unsolved Problems
You can contact Stephen C. Locke at LockeS@fau.edu Several people have asked me about unsolved problems. I will take the easy way out: see the list of 50 problems in Bondy and Murty . I hope it will not annoy the authors of that text if I will reproduce that list here, and perhaps (eventually) add to it. Problems number above 50 are from other sources. Some of these problems have been solved (and thus the title is slightly incorrect) and I won't claim to be familiar with all current results. If you find that one of them has been solved (or even that some reasonable progress has been made), please e-mail me . Also, I'm not giving you all of the references in Bondy and Murty . You should get yourself a copy of that book.
Problems 26-56
Problems 57-61 The reconstruction conjecture . (S.M. Ulam, 1960) 2. A graph G is embeddable in a graph H if G is isomorphic to a subgraph of H . Characterise the graphs embeddable in the k -cube. (V.V. Firsov, 1965) 3. Prove: Every 4-regular simple graph contains a 3-regular subgraph. (N. Sauer, 1973)
Conjecture 3 was proved in 1985 by L. Zhang: Every 4-regular simple graph contains a 3-regular subgraph, J. of Changsha Railway Institute 1 (1985), 130-154.

5. Unsolved Problems In Function Theory
Notes by Alexandre Eremenko.
http://www.math.purdue.edu/~eremenko/uns.html
My favorite unsolved problems GEOMETRIC FUNCTION THEORY AND POTENTIAL THEORY: ps pdf Some constants studied by Littlewood (Updated Oct 2002).
ps
pdf Exceptional set in Gross' Theorem.
ps
pdf "Hawaii Conjecture" (attributed to Gauss).
ps
pdf Does every universe contain a place where you can stay at rest? (Lee Rubel)
ps
pdf Erdos' problem on the length of lemniscates (at least \$200 prize). DIFFERENTIAL EQUATIONS AND ITERATION IN THE COMPLEX DOMAIN: ps pdf Wandering domains of entire functions. TRANSCENDENTAL HOLOMORPHIC CURVES: ps pdf Modified Cartan's Conjecture.
ps
pdf Holomorphic curves with few inflection points. RATIONAL FUNCTIONS AND RATIONAL CURVES: ps pdf Rational curves with real inflection points
##### Other interesting items in this site:
Progress report on some problems from Hayman's Collection When exactly had function theory became a secondary subject?
(Excerpt from a letter of Mittag-Leffler to Kowalevski.) What is mathematics? Some expert's opinions. jokes related to complex analysis some problems, whose solutions I do know (level: undergraduate+)

6. Unsolved Problems
Satements of some famous problems compiled by Frank Wikstrom, Ume¥ University.
http://abel.math.umu.se/~frankw/unsolved.html
 Mathematical research often concerns questions which are so abstract and technically complicated that only a small number of experts in that particular field can fully understand the problems and their significance. Therefore one is easily led to the conclusion that every problem is difficult to understand. This is not so. On this page I have gathered a handful of problems which to my knowledge and to this date are still unsolved although they can be understood by any one who has a working knowledge of, say, the high-school mathematics curriculum. Can every even integer number greater be written as the sum of two prime numbers? We have for example that: and so on, but is it true for every even number? Nobody knows, but most mathematicians seem to think that it is true. (The conjecture is known to be true for all even integers less than 20 000 000 000 or so. It is also known that every "sufficiently large" even integer can be written as the sum of a prime number and an integer with at most two prime factors.) Two consecutive odd numbers which are both prime are called twin primes, e.g. 5 and 7, or 41 and 43, or 1,000,000,000,061 and 1,000,000,000,063. But is there an infinite number of twin primes?

7. Unsolved Problems
the things that turned me on to math were some simple sounding but unsolved problems that were easy for a high school
http://www.math.utah.edu/~alfeld/math/conjectures.html
Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah
##### Some Simple Unsolved Problems
One of the things that turned me on to math were some simple sounding but unsolved problems that were easy for a high school student to understand. This page lists some of them.
##### Prime Number Problems
To understand them you need to understand the concept of a prime number A prime number is a natural number greater than 1 that can be divided evenly only by 1 and itself. Thus the first few prime numbers are You can see a longer list of prime numbers if you like.
##### The Goldbach Conjecture.
Named after the number theorist Christian Goldbach (1690-1764). The problem: is it possible to write every even number greater than 2 as the sum of two primes? The conjecture says "yes", but nobody knows. You can explore the Goldbach conjecture interactively with the Prime Machine applet.

Alexandre Eremenko (Purdue University). Mainly in analysis.
http://www.math.purdue.edu/~eremenko/
##### Alexandre Eremenko
picture vita Mathematics Department, Purdue University
149 N. University Street
West Lafayette, IN 47907-2068
OFFICE: Math 612, HOURS: Mo and We, 2-3.
PHONE: (765)494-1975, FAX: (765)494-0548
EMAIL: eremenko@math.purdue.edu
Courses that I teach and do not teach

My favorite unsolved problems
How do I choose journals to publish my papers Papers and Recent preprints (available in ps and pdf format) CO-AUTHORS: A. Atzmon, I. N. Baker, W. Bergweiler (3), V. Boichuk, M. Bonk (4), J. Clunie, N. Eremenko, B. Fuglede, A. Gabrielov (5), Yu. Gaida, A. A. Goldberg (6), D. Hamilton, W. Hayman J. Langley (2), L. Lempert, S. Merenkov D. Novikov , G. Levin (3), J. Lewis , T. Lyons, M. Lyubich (5), I. Ostrovskii (3), M. Ostrovskii, M. Petrika, J. Rossi (2), L. Rubel (2), D. Shea, M. Sodin (16). OTHER SITES
• MAG journal (Kharkov)
• Joseph Fourier
• Tom Korner
• Curt McMullen
• John Milnor ...
• XXX archive (Los Alamos)
• Jahrbuch uber die Fortschritte der Mathematik
• 9. Sci.math FAQ: Unsolved Problems
From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject sci.math FAQ unsolved problems Summary Part 18 of many,
http://www.faqs.org/faqs/sci-math-faq/unsolvedproblems
##### sci.math FAQ: Unsolved Problems
Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76LD.Fnt@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca By Archive-name By Author ... Help
Send corrections/additions to the FAQ Maintainer:
alopez-o@neumann.uwaterloo.ca
Last Update March 05 2003 @ 01:20 AM

10. Some Unsolved Problems
Mainly in analysis. By J¶rg Winkelmann.
http://www.math.unibas.ch/~winkel/problem.html
##### Some Mathematical Problems
This is a collection of some mathematical questions, which I encountered somehow, mostly in the context of my own research. To my knowledge they are all unsolved. Of course, this is not a complete list of all unsolved problems in mathematics. For instance, I omitted those problems which everybody knows anyway (like the Jacobi conjecture). Furthermore the choice is made following my personal taste and prejudices. I certainly do not want to claim that these are the most important problems in today mathematics. Nevertheless, I am very curious about the problems listed below. If anybody is able to solve one or more or knows some results in these directions, please tell me.
• Does there exists a compact Riemann surface M of genus at least two which can be embedded into a quotient of SL C by a discrete cocompact subgroup? (This is a question raised by A.T.Huckleberry.) Remark: This question is discussed in my Book on parallelizable manifolds and some partial results are derived. In particular, such a curve can never arise as a zero-section of a rank two vector bundle.
• Let S be a complex semisimple Lie group
• 11. Unsolved Problems
Puzzles and math Miscellany. unsolved problems. unsolved problems. unsolved problems in Operations Research. unsolved
 Math Problems, Games, and Puzzles Algorithmic Information Theory Brain Teasers Fermi Questions Library Ideas, Concepts, and Definitions ... What Good is Math?

12. Open Problems List
A collection of papers outlining unsolved problems in the field of dynamical systems.
http://www.math.sunysb.edu/dynamics/open.html
##### Open Problems in Dynamical Systems
We are soliciting open problems in various areas of Dynamical Systems for posting on this page. You can post a problem by filling out this form or by sending an e-mail to webmaster@math.sunysb.edu

13. Unsolved Problems (Part 2)
53 54 55 56 problems 125, problems 57-61 to me by Christopher Heckman, checkman@math.gatech.edu simplifiedproof of the 4-Color Theorem ( unsolved Problem 37
http://www.math.fau.edu/locke/unsolv2.htm
##### Unsolved Problems (Part 2)
You can contact Stephen C. Locke at LockeS@fau.edu Several people have asked me about unsolved problems. I will take the easy way out: see the list of 50 problems in Bondy and Murty . I hope it will not annoy the authors of that text if I will reproduce that list here, and perhaps (eventually) add to it. Problems number above 50 are from other sources. Some of these problems have been solved (and thus the title is slightly incorrect) and I won't claim to be familiar with all current results. If you find that one of them has been solved (or even that some reasonable progress has been made), please e-mail me . Also, I'm not giving you all of the references in Bondy and Murty . You should get yourself a copy of that book.
Problems 1-25
Problems 57-61 26. Prove that every n -chromatic graph G has r(G,G) >= r(n,n)
Message from Stephan Brandt
Disproved by Faudree and McKay [ J. Comb. Math. Comb. Comp. 13
Counterexample: Wheel with 5 spokes has Ramsey number 17 < r(4,4).

14. MATH-abundance
http://www.ping.be/math
##### MATH-abundance
• Main Purpose = MATH TUTORIAL
• TUTORIAL : Topics; Solved and unsolved problems
Last update: 29 jul 2002
##### Main Purpose = MATH TUTORIAL
The main purpose of this site is to provide the net with a 'upper secondary' MATH TUTORIAL The order of the topics is not random.
Most of them appeal on the properties and formulas stated in a preceding topic.
Since each topic is in 1 file, it is easy to download the file and to study the subject off line.
See Copying Conditions
##### Other Tutorials
Lessons, Tutorials and Lecture Notes

15. Www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems
math.uwaterloo.ca!neumann.uwaterloo.ca!alopezo From alopez-o@neumann.uwaterloo.ca(Alex Lopez-Ortiz) Subject sci.math FAQ unsolved problems Summary Part
http://www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems
 Newsgroups: sci.math,sci.answers,news.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!spool.mu.edu!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: Unsolved Problems Summary: Part 18 of many, New version, Originator: alopez-o@neumann.uwaterloo.ca Message-ID:

16. Mathsoft: Mathsoft Unsolved Problems
_product registration. mathsoft Resources. Statistics Resources. UnsolvedProblems. Engineering Standards. mathsoft Constants. math Resources. Welcome!
http://www.mathsoft.com/mathresources/problems/0,,0,00.html
 search site map about us  + news  + ... Statistics Resources Unsolved Problems Engineering Standards Mathsoft Constants Math Resources Welcome! This evolving collection of unsolved mathematics problems is not systematic or complete; it is only an eclectic gathering of questions and partial answers which have come to my attention over the years. web store ... contact us

17. Mathsoft: Mathsoft Unsolved Problems: Unsolved Problems On Other Sites
math Problem of the Week column and General References; Twin Primes Conjectureand Goldbach's Conjecture, discussed in Prime Numbers Some unsolved problems
http://www.mathsoft.com/mathresources/problems/article/0,,1999,00.html
 search site map about us  + news  + ... Unsolved Problems Unsolved Problems Links Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Engineering Standards Mathsoft Constants ... Math Resources Unsolved Problems on Other Sites Jeff Lagarias' 3x+1 problem and related problems The Generalized 3x+1 Mapping (University of Queensland) and 1999 Conference on the Collatz Problem Proceedings Alex Lopez-Ortiz's sci.math FAQ on Famous Problems in Mathematics (University of New Brunswick) Chris Caldwell's Riemann Hypothesis (University of Tennessee at Martin); also Daniel Bump's Riemann Hypothesis (Stanford University) and Barry Cipra's A Prime Case of Chaos MathPro Press Unsolved Math Problem of the Week column and General References Twin Primes Conjecture and Goldbach's Conjecture, discussed in Prime Numbers: Some Unsolved Problems , part of MacTutor History of Mathematics (University of St Andrews); also Chris Caldwell's Prime Conjectures and Open Questions (University of Tennessee at Martin) Jan Otto Munch Pedersen's Known Amicable Pairs (Vejle Business College, Denmark) and Chris Caldwell's

18. Understanding Mathematics
For our purposes it suffices to think of elementary school math as the main thingthat keeps mathematics alive and interesting of course are unsolved problems.
http://www.math.utah.edu/~alfeld/math.html
Peter Alfeld, Department of Mathematics, College of Science University of Utah
##### a study guide by Peter Alfeld.
I wrote this page for students at the University of Utah. You may find it useful whoever you are, and you are welcome to use it, but I'm going to assume that you are such a student (probably an undergraduate), and I'll sometimes pretend I'm talking to you while you are taking a class from me. Let's start by me asking you some questions. If you are interested in some suggestions, comments, and elaborations, click on the Comments. Do so in particular if you answered "Yes!". (In making the comments I assume you did say "Yes!", so don't be offended if you didn't and are just curious.)
• Do you feel
• That being lost in mathematics is the natural state of things?

19. Sci.math FAQ: Unsolved Problems
Subject sci.math FAQ unsolved problems. This article was archivedaround 17 Feb 2000 225551 GMT All FAQs in Directory scimath
http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/unsolved.html
Note from archivist@cs.uu.nl : This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archivist.