home | index | units | counting | geometry | algebra | trigonometry & functions | calculus
analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

Open Questions
or Tough Answers

© 2000-2016   Gérard P. Michon, Ph.D.

It is not because things are difficult
that we do not dare,  it is because
we do not dare that they are difficult.

Lucius Annæus Seneca   (5 BC - AD 65) 
As our island of knowledge grows,
so does the shore of our ignorance
John Wheeler   (Scientific American, 1992) 

The famous problems discussed below are either still unsolved or remained open for quite a while, in spite of considerable scrutiny.  Some entail substantial cash rewards.

Open questions mentioned elsewhere on this site:

Related articles on this site:

Related Links (Outside this Site)

What's New?  by Terence Tao.
Open Questions in Physics by John Baez.
MathPages Wanted List by Kevin S. Brown.
Proposed Proofs of the Riemann Hypothesis, by Matthew R. Watkins.
Dr. Grigory "Grisha" Perelman's 2002 proof of the  Poincaré conjecture
(generalized as  Thurston's geometrisation conjecture).
Open Problem Garden
Dark Buzz:  Mathematicians care about proofs  by  Roger Schlafly.

Cash Rewards for Answering Open Questions:

Millenium Prize Problems ($1000 000 each, from the Clay Mathematics Institute).
Million-Dollar MinesweeperLandon T. Clay's prize for the "P = NP" question.
John Baez's Description of the Millenium Prize Problems.
Origin-of-Life PrizeWhoever explains abiogenesis gets $67500 annuity (20 years).
$1,000,000 Unsolved Mathematical Problems for K-12  by  Gordon Hamilton.

Videos :

Mathematical Mystery Tour (BBC Horizon)
Some simple open problems in Mathematics (2013)  Joseph Oesterlé,  Paris VI.


Open Problems
(or recently resolved long-standing ones)

(2002-12-07)   RH  =  Riemann's Hypothesis   (August 1859)
The real part of all  nontrivial zeroes  of Riemann's z function is ½.

Without doubt it would be desirable to have a rigorous proof of this proposition.
However, I have left this research aside for the time being after some quick unsuccessful
attempts, because it appears to be unnecessary for the immediate goal of my study...

 Bernhard Riemann, 1859.

As he solved the  Basel problem  in 1735, Leonhard Euler (1707-1783) introduced the following series, now universally called Zeta, focusing on integral values of  s > 1  (for which the series converges).  Leonhard Euler 

z(s)   =   1  +  1 / 2s  +  1 / 3s  +  1 / 4s  +   ...   +  1 / ns  +   ...

In 1737, Euler observed that  (by the  fundamental theorem of arithmetic  which states that every positive integer has a  unique  factorization into primes)  the term of rank  n  in the above series is obtained once and only once in the expansion of the product of all the following  geometric series  for any set of primes containing, at least, every prime divisor  p  of  n.

1  +  1 / ps  +  1 / p2s  +  1 / p3s  +  ...  +  1 / pns  +  ...       =     1 / ( 1 - 1 / p)

So, the sum of the whole series is equal to the product of all such things for  all  primes  p.  This yields the following celebrated  Euler product formula :

z(s)   =   P p prime   ( 1 - p - s ) -1

 Charles de la Vallee-Poussin  
 (1866-1962) That formula  characterizes  the set of all prime numbers  (it isn't true for any other set of integers).  It's been the usual starting point of modern attacks on the  set of primes,  including the  simultaneous  proofs  (1896)  of the prime number theorem  (PNT)   by  Jacques Hadamard  (1865-1963)  and  Charles de la Vallée-Poussin  (1866-1962).

Both the series and the infinite product converge for any complex number  s  in the half-plane where the real part of  s  is greater than  1  (Re(s) > 1).

Consider now the related  Dirichlet eta function,  defined by the following  alternating series  which converges on the  right half-plane  (Re(s) > 0).

h(s)   =   1  -  1 / 2s  +  1 / 3s  -  1 / 4s  +   ...   +  (-1)n+1/ ns  +   ...

This is clearly equal to   z(s) - 2 z(s) / 2s .   Therefore:

z(s)   =   h(s) / (1 - 21-s )

Except for  s = 1,  this provides directly an analytic expression for  z(s)  in terms of an alternating series that converges when  Re(s) > 0.  Nice...

Dirichlet  used the above to show that  z  has a simple pole of  residue  1  at  s = 1.  (HINT:  The numerator and the denominator are respectively asymptotic to  Log 2  and  (s-1) Log 2  as  s  goes to  1.)

In 1859, Bernhard Riemann (1826-1866)  showed that the Zeta function can actually be extended to the entire complex plane, except at the  pole  s = 1,  by  analytic continuation  (a concept invented by  Weierstrass in 1842).  establishing in the process a relation between values at  (1-s)  and  s  (which had been conjectured by Euler in 1749, in an equivalent form):

p-½ s  G( s)  z(s)     =     p-½(1-s)  G( 1-s)  z(1-s)
Vinculum Vinculum

Using the known properties of the  Gamma function  (whose reciprocal  1/G  is an  entire function  with zeroes at all the nonpositive integers)  this relation confirms the existence of a simple pole for the  Zeta function  at  s = 1  and reveals its so-called  trivial zeroes  at  -2, -4, -6, etc.

Using the regularity of the  Zeta function  for  Re(s) > 1  (due to the aforementioned convergence of the defining series in that domain)  this same relation shows that  nontrivial zeroes  can only exist in the so-called  critical strip  (0 ≤ Re(s) ≤1).  They could thus  a priori  be of two different types:

  • Pairs of zeroes  { s, s* }  on the  critical line  (i.e., Re(s) = ½).
  • Quadruplets of zeroes  { s, s*, 1-s, 1-s* )  off that  critical line.

The famous  Riemann Hypothesis  (RH)  is the  conjecture,  formulated by Riemann in 1859,  that there are no nontrivial zeroes of the latter type:

"RH"   The Riemann Hypothesis   (1859)
All nontrivial zeroes of the Zeta function are on the critical line :   Re(s) = ½

Zeta shares its nontrivial zeros with the above convergent series:  (Re(s)>0)

h(s)   =   1  -  1 / 2s  +  1 / 3s  -  1 / 4s  +   ...   +  (-1)n+1/ ns  +   ...

Listed below are the imaginary parts of the  29  smallest zeroes of the  Zeta function  located in the upper half-plane  (conjugate zeroes exist in the  lower half-plane  whose imaginary parts are simply the opposites of these).  The gigantic  ZetaGrid  distributed project of  Sebastian Wedeniwski  managed to compute more than  10 trillion  zeros over the course of its lifetime  (2001-2005)  but they were scooped by  Xavier Gourdon and Patrick Demichel  who achieved that same goal earlier with modest means by using superior software based on an algorithm by Odlyzko and Schönhage...

14.1347251417346937904572519835624702707842571156992431756855674+ 21.0220396387715549926284795938969027773343405249027817546295204+ 25.0108575801456887632137909925628218186595496725579966724965420+ 30.4248761258595132103118975305840913201815600237154401809621460+ 32.9350615877391896906623689640749034888127156035170390092800034+ 37.5861781588256712572177634807053328214055973508307932183330011+ 40.9187190121474951873981269146332543957261659627772795361613037- 43.3270732809149995194961221654068057826456683718368714468788937- 48.0051508811671597279424727494275160416868440011444251177753125+ 49.7738324776723021819167846785637240577231782996766621007819558- 52.9703214777144606441472966088809900638250178888212247799007481+ 56.4462476970633948043677594767061275527822644717166318454509698+ 59.3470440026023530796536486749922190310987728064666696981224518- 60.8317785246098098442599018245240038029100904512191782571013488+ 65.1125440480816066608750542531837050293481492951667224059665011- 67.0798105294941737144788288965222167701071449517455588741966696- 69.5464017111739792529268575265547384430124742096025101573245400- 72.0671576744819075825221079698261683904809066214566970866833062- 75.7046906990839331683269167620303459228119035306974003016477753+ 77.1448400688748053726826648563046370157960324492344610417652315- 79.3373750202493679227635928771162281906132467431200308784387205- 82.9103808540860301831648374947706094975088805937821491465713063- 84.7354929805170501057353112068277414171066279342408187027355297- 87.4252746131252294065316678509192132521718864012690281864555579+ 88.8091112076344654236823480795093783954448934098186750421998716+ 92.4918992705584842962597252418106848787217940277306461750967505- 94.6513440405198869665979258152081539377280270156548520195924743- 95.8706342282453097587410292192467816952564612249879984205292817- 98.8311942181936922333244201386223278206580390634281961028193217+

Statements Equivalent to the Riemann Hypothesis :

Many statements have been shown to hold if RH is assumed to be true, and a number of them are known to imply RH, so they are actually equivalent to it.

In 1901, Helge von Koch showed that RH is equivalent to a tight bound between the  prime counting function  p(x)  and the Logarithm integral :  Helge von Koch 

p(x)   =   Li (x)  +  O ( x½ Log x )

Beyond the Riemann Hypothesis :

Several nice statements have been made which  seem  true and are simpler and stronger than RH (each implies RH but the converse need not be true).  This includes a conjecture made in the doctoral dissertation of  Sebastian Wedeniwski  (the aforementioned mastermind of  ZetaGrid,  2001-2005) :

Wedeniwski's Property   (2001)
Modulo p,  there is a  quadratic nonresidue  below   3/2 (Log p)2

Riemann Hypothesis at...   Wikipedia   |   MathWorld   |   Prime Pages   |   Clay Mathematics Institute
How Euler discovered the zeta function   by  Keith Devlin  (Devlin's Angle at MAA, The Math Guy at NPR, etc.)
Euler and the Zeta Function  by  Sumida-Takahashi Hiroki   (Hiroshima University)
Zeta function   |   Eta function   |   Ten Trillion Zeta Zeros   by  Ed Pegg, Jr.
Videos :   The Riemann Hypothesis  by  James Grime  "Singingbanana"  (2014).
The Riemann Hypothesis  by  Alex Kontorovich  (Math Mornings at Yale, 2012-12-02).

To the tune of  Sweet Betsy from Pike  (c.1858  by  "Old Put" = John A. Stone)
The Zeta Function Ballad  (1955)
Tom Mike Apostol<opoulos> (1923-2016)
 Additional modern verses (1973)
by Saunders Mac Lane (1909-2005)
Where are the zeros of zeta of s ?
G.F.B. Riemann has made a good guess:
" They're all on the critical line," said he.
" And their density's one over two pi Log t. "
This statement of Riemann's has been like a trigger
And many good men, with vim and with vigor,
Have attempted to find, with mathematical rigor,
what happens to zeta as mod t gets bigger.
The names of Landau and Bohr and of Cramer,
Hardy and Littlewood, and Titchmarsh are there.
In spite of their efforts and skill and finesse,
in locating the zeros there's been no success.
In 1914, G.H. Hardy did find,
An infinite number that lay on the line,
His theorem however won't rule out the case,
That there might be a zero at some other place.
Let P be the function pi minus Li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann's conjecture would surely be so.
Related to this is another enigma,
Concerning the Lindelhöf function mu(sigma)
Which measures the growth in the critical strip,
On the number of zeros it gives us a grip.
But nobody knows how this function behaves.
Convexity tells us it can have no waves,
Lindelhof said that the shape of its graph,
Is constant when sigma is more than one-half.
Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess,
In order to strengthen the prime number theorem,
The integral's contour must not get too near 'em.
André Weil has bettered old Riemann's fine guess,
by using a fancier zeta of s.
He proves that the zeros are where they should be,
provided the characteristic is p.
There's a moral to draw from this sad tale of woe.
That every young genius among you should know:
If you tackle a problem and seem to get stuck,
just take it mod p and you'll have better luck.
What fraction of zeros on the line will be found,
when mod t is kept below some given bound?
Does the fraction, whatever, stay bounded below,
as the bound on mod t is permitted to grow?
The efforts of Selberg did finally banish,
All fears that the fraction might possibly vanish.
It stays bounded below, which is just as it should,
But the bound he determined was not very good.
Norm Levinson managed to show, better yet.
At two-to-one odds it would be a good bet,
If over a zero you happen to trip,
it would be on the line and not just in the strip.
Levinson tried in a classical way,
Weil brought modular means into play.
Atiyah then left and Paul Cohen quit,
So now there's no proof at all that will fit.
But now we must study this matter anew,
Serre points out manifold things it makes true.
A medal might be the reward in this quest,
For Riemann's conjecture is surely the best.

The paper of Bernhard Riemann was published in November 1859  (Monatsberichte der Berliner Akademie):
Über die Anzahl der Primzahlen unter einer gegebenen Grösse  (German original, 1859)
On the Number of Primes less than a Given Quantity  (English translation, by David R. Wilkins, 1998)

(2013-07-02)   The Twin Primes Conjecture
Are there infinitely many pairs of primes whose difference is  2 ?

The first such pairs are:   {3,5}  {5,7}  {11,13}  {17,19}  {29,31} ...

At this writing,  the largest known pair of twin primes  (200700 digits)  was discovered by  Timothy D. Winslow,  PrimeGrid, et al. in December 2011:

{ 3756801695685 . 2666669 - 1   ,   3756801695685 . 2666669 + 1 }

Nobody knows for sure if that sequence is infinite, although everybody is guessing that it is.  This statement is one of the two  oldest  unsolved problems in mathematics  (the other one pertains to  odd perfect numbers).

The  Twin Primes Conjecture  says that the following is true for  K = 2:

There are infinitely many pairs of primes whose difference is  K.

It's widely believed that the above statement holds for  any even integer  K. Polignac's coat-of-arms

Polignac's conjecture (1849) is the belief that there are infinitely many pairs of  consecutive  primes differing by  K, for any even K.

The weaker statement that the above holds for  at least one  nonzero value of  K  is equivalent to saying that the difference between consecutive primes doesn't tend to infinity.  This was only proved recently:

Yitang Zhang
Yitang "Tom" Zhang, U. of New Hampshire

  In April 2013,  Yitang Zhang  established an upper-bound of  70 million  for the least  K  verifying the above.
Zhang was quoted as saying that the methods in his 55-page paper could be perfected to pull this upper-bound downward.
Quickly,  Terry Tao  initiated a polymath project  which reduced the upper-bound to 4680, using Zhang's own methods. 

In November 2013,  James Maynard  (b. 1987)  proposed a streamlined approach which gave directly an upper-bound of  600.  He joined forces with Tao's group and they finally managed to reduce the bound to 246  (as of  2014-04-14).

Assuming a generalized version of the  Elliott-Halberstam conjecture (1968) the above lower-bound would be reduced down to 6  [Polymath, August 2014].  However, new methods seem needed for the ultimate reduction to 2.

"Bounded Gaps between Primes"  by Yitang Zhang  (Annals of Mathematics, 179, pp.1121-1174)
Zhang's Theorem on Bounded Gaps between Primes  by  Dan Goldston (1954-).
First proof that infinitely many primes come in pairs  by  Maggie McKee  (Nature, 2013-05-14)
Unheralded Mathematician Bridges the Prime Gap  by  Erica Klarreich   (Simons Foundation, 2013-05-19)
Philosophy behind Yitang Zhang's work  (MathOverflow, 2013-05-20)
Videos :   Gaps between Primes   &  extra footage,  in  "Numberphile"  by  Brady Haran  (2013-05-27)
Terry Tao, Ph.D.   Small and Large Gaps Between the Primes  (UCLA, 2014-10-07)
Yitang Zhang's Counting From Infinity Bonus Material :   1 | 2 | 3 | 4 | 5 | 6 | 7

(2016-01-29)   Golbach Conjectures   (1742)
Every even number greater than  2  is the sum of two primes.

That conjecture was first formulated by the talented recreational mathematician  Christian Goldbach (1690-1764)  who wrote to  Euler  about it in 1742.

An equivalent satement is obtained with odd primes by excluding the number  4 = 2+2  (which isn't the sum of two odd primes).

The weaker statement that odd numbers are sums of three primes can be construed as a corollary, from the remark that an odd number above 5 is 3 plus an even number above 2.  That weaker statement is less formidable; it was shown to hold for sufficiently large odd numbers in 1923 by  Hardy & Littlewood  assuming  the  Riemann Hypothesis.

  • Even Goldbach Conjecture :   (Goldbach's conjecture)
    All even numbers above 2 are sums of two primes.
    All even numbers above 4 are sums of two odd primes.
  • Odd Goldbach Conjecture :   (Goldbach's weak conjecture)
    All odd numbers above 5 are sums of three primes.
    All odd numbers above 7 are sums of three odd primes.

In 2012 and 2013, Harald Helfgott (b. 1977)  published two papers which provide a complete proof of the weak conjecture.

The strong conjecture remains an open question which has only been checked by computer for even numbers up to  4 1018  or so.

On 1752-11-18,  Goldbach also formulated the lesser-known conjecture that any odd number is twice a square plus a prime.  Two counterexamples  (5777 and 5993)  were discovered in 1856 by  Moritz A. Stern  (of  Stern-Brocot tree fame).

Wikipedia:   Goldbach's (strong) conjecture   |   Goldbach's weak conjecture

(2002-11-17)   P = NP (?)
What's an NP-complete problem?

A computational problem is said to be in the class P of polynomial time problems whenever there's an algorithm which can find a valid solution in a number of elementary steps which is always less than a certain  polynomial function  of the  size  of the input data  (one measure of this size could be the number of digits used in a reasonnably thrift encoding of the input data).

The class NP (nondeterministic polynomial time problems) consists of those problems which could be so solved  nondeterministically, which is a fancy way to say that an unlimited number of lucky guesses are allowed in the process which arrives at a solution.  Such a nondeterministic process must still be such that only valid solutions are produced...  To put it in simpler words, a problem is in NP if and only if a solution of it can be  checked  in polynomial time  (an explicit nondeterministic algorithm would then be to guess a correct solution and check it).

In 1972, Richard Karp (1935-)  discovered that there are problems in NP which he dubbed "NP-complete" because they are at least as tough to solve as any other problem in NP, in the following sense:  Any NP problem can be reduced in polynomial time by a deterministic algorithm to the solution of an NP-complete problem whose data size is no more than a polynomial function of the original input data.

Therefore, if any NP-complete problem could be solved deterministically in polynomial time (i.e., if it was a P problem) then  all  NP problems would be in P and we would thus have  P = NP.

Karp's original NP-complete problem (dubbed SAT) was the satisfiability of boolean expressions:  Is there a way to  satisfy  a given boolean expression (i.e., make it "true") by assigning true/false values to the variables in it?

The SAT problem is clearly in NP (just guess a correct set of values and compute the boolean expression to make sure it's true).  Conversely, Stephen Cook (1939-)  proved from scratch in 1971 that any problem in NP can be reduced in polynomial time to a commensurable boolean satisfiability problem, thus establishing SAT to be NP-complete.  This result is now known as the Cook-Levin theorem because it was also obtained independently by  Leonid Levin (1948-).

If a known NP-complete problem like SAT can be reduced polynomially to some NP problem Q, the problem Q is then established to be NP-complete.  This way, from Karp's original NP-complete problem, the list of known NP-complete problems has grown to include literally hundreds of "classical" examples.

The tantalizing thing is that many such NP-complete problems are very practical problems which, at first, look like they could be solved in polynomial time.  Yet, nobody has ever "cracked" one of these in polynomial time or proved that such a thing could not be done...  Therefore we still don't know whether P=NP or not.

(2002-12-21)   The Collatz Problem
Collatz sequences are sequences of integers where an even N is followed by N/2 and an odd N by 3N+1.  Do they all end up with  4, 2, 1, 4...?

The problem is most commonly named after the German mathematician  Lothar Collatz  (1910-1990)  who formulated the conjecture in 1937.

It's also called the hailstone problem  (because of its dynamics)  and has also been popularized under several other names at times when the thing became more prominently associated with a particular individual or a specific institution:  The Syracuse problem  (Syracuse University).  Kakutani's problem  (Yale University, early 1960's).  Ulam's problem.

 Come back later, we're
 still working on this one...

On the Dirichlet inverse of the integers modulo 2   (A087003)

The sequence A087003,  may be defined as the Dirichlet inverse of the character modulo 2.  It was first defined by  Labos E.  as the sum of all the Möbius values found at the points of the Collatz trajectory until a "4" is found  (2003-10-02).

However, Marc Lebrun (2004-02-19) has shown that either definition simply means that A087003 is equal to the Möbius function at odd points and vanishes at even points...  All told, the Collatz trajectories turn out to be irrelevant !

Videos:   Norman J. Wildberger   (2013-01-14)
Marc Chamberland , Grinnell College :   Part 1  |  Part 2   (2013-04-12).

(2006-08-29)   The Poincaré Conjecture   (1904)
This was proved in 2002 by  Grigori Yakovlevich Perelman,  in the generalized form proposed by Thurston  (geometrization conjecture).

In 1904, Henri Poincaré had conjectured that:

The  3-sphere  is the only closed 3-manifold in which
every loop can be continuously tightened to a point.

 Come back later, we're
 still working on this one...

Poincaré conjecture  |  Geometrization conjecture of William Thurston  |  Grisha Perelman (1966-)

(2010-10-15)   Fermat's Last Theorem   (Fermat 1637, Wiles 1995)
xn + yn = zn   has no solutions in positive integers for  n > 2.

 Pierre de Fermat  
 (1601-1665)  Hanc marginis exiguitas non caperet.  
This margin is too small to contain [my proof].  
Pierre de Fermat (1601-1665)  

Die Gleichung  an=bn+cn  für  n>2 in ganzen Zahlen  [ist]  nicht auflösbar.
A. Ernest Wendt  (1894)

Solutions for  n = 2  are called  Pythagorean triples.  They are fairly easy to enumerate systematically, starting with x=3, y=4, z=5.  Many special cases known in ancient times were recorded on  Chaldean clay tablets.

In the Middle Ages, Leonardo Fibonacci proved that there was no solutions for n = 4  (Liber Quadratorum, 1225).

 Come back later, we're
 still working on this one...

Fermat's Last Theorem, a film by  Simon Singh & John Lynch  (1996)
Nicholas M. Katz

(2013-01-09)   The  Oesterlé-Masser   ABC Conjecture
Moshizuki  has proposed a 500-page proof whose philosophy is obscure.

The  radical  (rad)  of a positive integer  n  is the integer whose  prime factorization  consist of the same primes as that of  n  with  multiplicity  1.  The function  rad  is a  multiplicative function.

The  ABC conjecture  says that the inequality   rad ( a b ) > (a+b)e   has infinitely many exceptions when  e = 0  but finitely many when  e > 0.

The conjecture was formulated in 1985 by the French  Bourbakist  Joseph Oesterlé (b. 1954)  and the British mathematician  David Masser (b. 1948).

That's arguably the most important open  Diophantine statement  today.

On August 30, 2012,  Shinichi Mochizuki, from Kyoto, released...

 Come back later, we're
 still working on this one...

The Mochizuki Theorem?   by  Jeremy Teitelbaum,  University of Connecticut  (2012-10-11).
Episode of "Numberphile" by Brady HaranJames Grime presents the ABC conjecture  (2012-10-12).
An ABC proof too tough to check   by  By Kevin Hartnett,  Boston Globe  (2012-11-03).
Wikipedia :   abc conjecture   |   David Masser (1948-)   |   Joseph Oesterlé (1954-)

(2016-05-30)   Union-closed sets conjecture   (Péter Frankl, 1979)
In a nontrivial finite  union-closed  family of finite sets,
is there always an element that belongs to at least half of the sets?

The "nontrivial" qualifier indicates that we're considering only families containing at least one nonempty set.  A family of sets is said to be  union-closed  when it contains any union of its members.

The conjecture clearly holds for families containing at least one singleton.

Wikipedia :   Union-closed sets conjecture (1979)   |   Péter Frankl (1953-)

(2016-05-30)   The Egyptian conjecture   (Straus & Erdös, 1948)
For n≥2,  is  4/n  always the sum of three  unit fractions?

Besides 2/3 and 3/4,  ancient Egyptians only allowed fractions with numerator 1  (unit fractions).  They represent other fractions as sums of those  (without repetitions).

Paul Erdös (1913-1996)   |   Ernst G. Straus (1922-1983)   |   Wikipedia :   Egyptian conjecture (1948)

visits since April 14, 2007
 (c) Copyright 2000-2016, Gerard P. Michon, Ph.D.