It is not because things are difficult that we do not dare, it is because we do not dare that they are difficult. Lucius AnnæusSeneca  (5 BC - AD 65)    As our island of knowledge grows, so doestheshore of our ignorance. John Wheeler  ( 1992)  The Riemann Hypothesis:   {Re(s) 0   &   (s) = 0}     {Re(s) = ½}. Twin prime conjecture:  One of theoldest open mathematical questions. Goldbach conjecture (1742):  Any even number  >2  is a sum of two primes. Legendre's conjecture  There are primes between squares. Landau's fourth problem;  Infinitely many primes follow a square. Schnizel's hypothesis H.  Polynomials simultaneously prime infinitely often. P = NP ?   Can we find  in polynomial time what we can check  that fast? Collatz sequences  go from  n  to  n/2  or  (3n+1)/2.  Do they all lead to  1? The Poincaré Conjecture (1904).  Proven by Grisha Perelman  in 2002. Fermat's Last Theorem (1637).  Proven by Andrew Wiles  in 1995. The ABC conjecture (1985)  of Joseph Oesterlé  and David Masser. The union-closed sets conjecture (1979)  of Péter Frankl. The Hadwiger conjecture in graph theory (1943). The Egyptian conjecture (1948)  of Straus  and Erdös. Birch and Swinnerton-Dyer conjecture  about the rank of an elliptic curve. Gilbreath's conjecture  was first formulated by François Proth  in 1878. g-conjecture  of McMullen  (1970).  Proved by Karim Adiprasito  in 2018. Positive C2-diffeomorphisms on the circle.  Do they form a simple  group? Hadwiger-Nelson colorings are such that 2 like points are never 1 unit apart. Tough questions mentioned elsewhere on this site: Continuum Hypothesis. Noncommutative Geometry. There are Hadamard matrices  of any order multiple of 4? Computer-aided proofs: 4-color theorem  (Appel & Haken, 1976). Odd Perfect Numbers:  One of theoldest unsolved mathematical questions. Ramsey's Theorem:  Monochromatic complete subgraphs of a large graph. Carmichael's conjecture:  There's a totient obtained for only one  number. Divisors of Carmichael numbers  form  the set of all odd cyclic numbers. Is the Euler-Mascheroni constant  ()  irrational?  Transcendental? J.J. Thomson's problem:  N  repelling charges on a spherical surface. Continued fraction expansion of theDelian constant:  Is isnormal? Artin's conjecture  about primitive roots. Related articles on this site: Parodies and hoaxes. Proof of Fermat's last theorem. Factorization of Wendt's determinant. Scientific Mysteries: The Unexplained. A solution looking for a problem  (Numericana, 2014-07-26). Cosmology 101:  Basic concepts for studying the Universe as a whole. Related Links (Outside this Site) Landau's 4 problems  related to prime numbers (1912) Open Questions in PhysicsbyJohn Baez. MathPages Wanted ListbyKevin S. Brown. Open Problem Garden  |  What's New? byTerence Tao. Dark Buzz: Mathematicians care about proofs  by Roger Schlafly. Thanks for Additivity (progress in knowledge)  by Kenneth W. Regan  (2015). Openproblems in number theory  by Michel Waldschmidt  (2017-03-07). Innocentive Millenium Prize Problems Million-DollarMinesweeper:  John Baez'sDescription of the Millenium Prize Problems. Origin-of-Life Prize:  $1,000,000 Unsolved MathematicalProblems for K-12  by Gordon Hamilton. Mathematical Mystery Tour (BBC Horizon) Some simple open problemsin Mathematics (2013) Joseph Oesterlé,  Paris VI. Four of the seven millenium problems by Michael Atiyah  (2000-05-24).

	Open Problems (2002-12-07)   The real part of all nontrivial zeroes  of Riemann's function is ½. I have temporarily put aside the search for a rigorous proof,  after some futile attempts,because it's not necessary for my immediate goal.  Bernhard Riemann, 1859. As he solved the Basel problem in 1735,Leonhard Euler (1707-1783)introduced the following series, now universally calledZeta,focusing on integral values of  s > 1  (for whichthe series converges). (s)   =  1  + 1/ 2s  + 1/ 3s  + 1/ 4s  +   ...   + 1/ ns  +   ... Using the fundamental theorem of arithmetic (which states that every positive integer has a unique  factorization into primes) Euler observed  (in 1737)  that the term of rank  n  in the above series is obtained once and only once in the expansionof 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/ ps ) 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 : (s)   =   p prime  ( 1 p s) 1 That formula characterizes  theset 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 theprime number theorem  (PNT)  by Jacques Hadamard  (1865-1963)  and Charles de la Vallée-Poussin  (1866-1962). Both the series and the infinite product convergefor anycomplex number  s in the half-plane where the real part of  s  is greater than  1 (Re(s) > 1). Consider now the related Dirichleteta function,  defined by the following alternating series which converges on the right half-plane (Re(s) > 0). (s)   =  1   1/ 2s  + 1/ 3s   1/ 4s  +   ...   + (-1)n+1/ ns  +   ... At least when  (s)  converges, that's equal to  (s) 2 (s)/ 2s .   So: (s)   =  (s)/ (1 21-s) Except for  s = 1,  this provides directly an analytic expression for (s)  in terms of an alternating serieswhich converges when  Re(s) > 0.  Nice...  has a simple pole of residue 1  at  s = 1.  (:  The numerator and thedenominator are respectively asymptotic to Log 2  and (s-1) Log 2  as  s  goes to  1.) In 1859,Bernhard Riemann (1826-1866)  showed thatthe Zeta function can actually be extended to the entire complex plane,except at the pole  s = 1, by  analytic continuation (a concept invented by Weierstrassin 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): -½ s  s  (s)    =    -½(1-s)  1-s  (1-s) 2 2 Using the known properties of the Gamma function (whose reciprocal  1/ is an entire function with zeroes at all the nonpositive integers)  this relation confirms the existence ofa simple pole for the Zeta function  at s = 1  and reveals trivial zeroes  at negative even integers:  -2, -4, -6... Using the regularity of the Zeta function  for  Re(s) > 1 (due to theaforementioned convergence of the defining series in that domain) this same relation shows that nontrivial zeroes  can only existin 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)  isthe conjecture,  formulated byBernhard Riemann in 1859, that there are no  zeroes of the latter type: "RH"  The Riemann Hypothesis  (1859) All nontrivial zeroes of the Zeta function are on the critical line :  Re(s) =  Zeta shares its nontrivial zeros with the above convergent series:  (Re(s)>0) (s)   =  1   1/ 2s  + 1/ 3s   1/ 4s  +   ...   + (-1)n+1/ ns  +   ... Listed below are the imaginary parts of the  29  smallest zeroesof the Zeta function located in theupper half-plane (conjugate zeroes exist in the lower half-plane whose imaginary parts are simply the opposites of these). The gigantic ZetaGrid  distributed projectof Sebastian Wedeniwski managed to compute more than  10 trillion zeros over the course of its lifetime  (2001-2005)  but they were scoopedby XavierGourdon and Patrick Demichel  who achieved that same goal earlier with modest means by usingsuperior software based on an algorithm devised in 1988 by Andrew M. Odlyzko (1949-)  and Arnold Schönhage (1934-)... 14.1347251417346937904572519835624702707842571156992431756855674+21.0220396387715549926284795938969027773343405249027817546295204+25.0108575801456887632137909925628218186595496725579966724965420+30.4248761258595132103118975305840913201815600237154401809621460+32.9350615877391896906623689640749034888127156035170390092800034+37.5861781588256712572177634807053328214055973508307932183330011+40.918719012147495187398126914633254395726165962777279536161303743.327073280914999519496122165406805782645668371836871446878893748.0051508811671597279424727494275160416868440011444251177753125+49.773832477672302181916784678563724057723178299676662100781955852.9703214777144606441472966088809900638250178888212247799007481+56.4462476970633948043677594767061275527822644717166318454509698+59.347044002602353079653648674992219031098772806466669698122451860.8317785246098098442599018245240038029100904512191782571013488+65.112544048081606660875054253183705029348149295166722405966501167.079810529494173714478828896522216770107144951745558874196669669.546401711173979252926857526554738443012474209602510157324540072.067157674481907582522107969826168390480906621456697086683306275.7046906990839331683269167620303459228119035306974003016477753+77.144840068874805372682664856304637015796032449234461041765231579.337375020249367922763592877116228190613246743120030878438720582.910380854086030183164837494770609497508880593782149146571306384.735492980517050105735311206827741417106627934240818702735529787.4252746131252294065316678509192132521718864012690281864555579+88.8091112076344654236823480795093783954448934098186750421998716+92.491899270558484296259725241810684878721794027730646175096750594.651344040519886966597925815208153937728027015654852019592474395.870634228245309758741029219246781695256461224987998420529281798.8311942181936922333244201386223278206580390634281961028193217+ Statements Equivalent to the Riemann Hypothesis : Many statements have been shown to hold if RH is assumed to be true, anda number of them are known to imply RH, so they are actually equivalent to it. In 1901,Helge von Koch(1870-1924)  proved RH equivalent to a relation betweenthe prime counting function (x) and thelogarithmic integral : (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 thanRH (each implies RH but the converse need not be true). This includes a conjecture made in the doctoral dissertation of Sebastian Wedeniwski  (theaforementioned mastermind of ZetaGrid,  2001-2005) : Wedeniwski's Property   (2001) Modulo p,  there is a quadratic nonresidue  below  3/2 (Log p)2

The Mertens conjecture (Stieltjes, 1885.  Mertens, 1897) was yet another famous statement "stronger than RH" which seemed promising until it was disproved in 1985by Andrew Odlyzko (b.1949)  and Herman te Riele (b.1947). It had been formulated in 1885 by Thomas Stieljes (1856-1894) after he thought he could prove the following weaker statement (actually equivalent to RH)  about the asymptotics  of the Mertens function  M :

M(n)   =   O(n ^½+ )      for any > 0

The Mertens conjecture  is the stronger (false) statement:  |M(n)|  <  n^½

(2013-07-02)  
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} ...

In December 2011, a large pair of twin primes  (200700 digits) was discovered by  Timothy D. Winslow,  PrimeGrid, et al. :

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

This remained the largest known until2016-09-14,when a pair of twin primes with 388342 digitswas found by  Tom N. Greer,  PrimeGrid, et al. :

{ 2996863034895 . 2^1290000 1  ,   2996863034895 . 2^1290000 + 1 }

Nobody knows for sure whether that sequence is infinite or not, although everybody's guessing that it is. That's one of the two oldest unsolved problems in mathematics  (the other one pertains to odd perfect numbers).

The Twin Primes Conjecture  says that thefollowing 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.

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

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 inhis 55-page papercould be perfected to pull this upper-bound downward.   Quickly, Terry Tao  initiated apolymathproject  which reduced the upper-bound to 4680,using Zhang's own methods.    In November 2013, James Maynard (b. 1987, 2002 Fields Medal) found a streamlined approach giving directly an upper-bound of  600. He joined 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-Halberstamconjecture (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.

(2016-01-29)  
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 construedas 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 sufficientlylarge 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 beenchecked by computerfor even numbers up to  4 ¹⁸  or so.

(2017-07-08)     (Legendre, 1808)
Is there always at least one prime between two nonzero perfect squares?

There seems to be at least two  primes between two consecutive squares.

A014085(n)  is the number of primes between n²  and  (n+1)².  Namely:

0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10,9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16,13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15...

Asymptotically, there should be as many primes between  n²  and  (n+1)² as between  1  and  n  (roughly n / ln n, by the prime-number theorem). So, we're very confident that Legendre's conjecture  won't failin the long run.  That's a goodheuristic argumentbut it doesn't constitute a proof.

(2017-07-31)  
Are there infinitely many primes following a square?

A005574  is the sequence of the numbers  n for which  n²+1  is prime:

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124,126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250,256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396,400, 406, 420, 430, 436, 440, 444, 464, 466, 470, 474, 490, 496,536, 544, 556, 570, 576, 584, 594...

It seems likely that there are infinitely many primes of this form but thisis not known for sure.  This old  questionwas popular enough in 1912 for Landau to include it in hislist of four unsolved problems related to primes.

(2017-11-25)    (Schinzel &Sierpinski, 1958)
A set of irreducible integer-valued polynomials without a fixed prime divisorare simultaneously prime infinitely often.

Note that,  bt definition,  an irreducible polynomial is non-constant.

Integer-valued polynomials need not have integer coefficients.
½ n² + ½ n  is integer-valued  (A000217)  but it's reducible.

A fixed divisor  of several integer-valued polynomials isdefined to be an integer which divides their product at every point. A set of polynomials without a prime fixed divisor  is said to verify Bunyakovsky's property.

Bunyakovsky's property  is clearly necessary for nonconstantpolynomials to be simultaneously prime infinitely often. Otherwise, there would be a fixed prime  p dividing the product of the polynomials at every point...

In this case, at every point where all polynomials are prime, at least one ofthem must be equal to  p  (if a prime divides a product of primes,it's equal to one of them).

This implies that at least one of the polynomials is equal to  p infinitely many times,  which can only happen if it's constant,  which is ruled out. 

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

A computational problem is said to be in the class Pofpolynomial time problems whenever there'sanalgorithm which can find a valid solution ina number of elementary steps which is always less than a certain polynomial function of the size  of the input data  (one measure of thissize could be the number of digits used in a reasonnably thrift encoding ofthe 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 allowedin the process which arrives at a solution. Such a nondeterministic process must still be such that only valid solutionsare produced... To put it in simpler words, a problem is in NP if and only if a solution ofit can be checked  in polynomial time (an explicit nondeterministic algorithm would then be to guess a correct solutionand 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 thefollowing sense: Any NP problem can be reduced in polynomial time by a deterministic algorithmto the solution of an NP-complete problem whose data size is no more thana polynomial function of the original input data.

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

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

The SAT problem is clearly in NP (just guessa correct set of values and compute the booleanexpression to make sure it's true). Conversely,Stephen Cook (1939-) proved from scratch in 1971 that any problem in NP can be reduced in polynomialtime to a commensurable boolean satisfiability problem, thus establishingSAT to be NP-complete.  This result is now known as theCook-Levin theorembecause 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 includeliterallyhundreds of "classical" examples.

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

(2002-12-21)  
Collatz sequences are sequences of integers where an even N is followed by N/2and an odd N by 3N+1. Do theyall land on  4, 2, 1, 4...?

The problem is most commonly named after the German mathematician LotharCollatz  (1910-1990)  who formulated the conjecture in 1937 and shared it privately withStanislaw Ulam (1909-1984)andShizuo Kakutani (1911-2004)at theICM in 1950.

This was first described in print in 1971, in a transcript of atalk given in 1970 byHarold Coxeter (1907-2003). In 1975,Helmut Hasse (1898-1979)   coined the name Syracuse problem  after Syracuse University.

On theDirichlet inverse of theintegers modulo 2 (A087003)

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

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

Breakthrough in 2019

Almost all Collatz orbits attain almost bounded values.

This is about as close as anyone can get to the
Collatz comjecture without actually solving it.
Terry Tao (September 2019)

(2006-08-29)  
This was proved in 2002 by Grigori Yakovlevich Perelman, in the generalized form proposed byThurston  (geometrization conjecture).

In 1904,Henri Poincaré had conjectured that:

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

(2010-10-15)  
xⁿ + yⁿ = zⁿ  has no solutions in positive integers for  n > 2.

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 Pythagoreantriples.  They are fairly easy to enumerate systematically, starting withx=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 forn = 4  (LiberQuadratorum, 1225).

(2013-01-09)  
Moshizuki  has proposed a 500-page proof whose philosophy is obscure.

The radical  (rad)  of a positive integer  n  is theinteger whose prime factorization  consist of thesame 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)  has infinitely many exceptions when   = 0 but finitely many when   > 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...

(2016-05-30)  
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 onenonempty set.  A family of sets is said to be union-closed  when itcontains anyunion of its members.

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

(2019-07-18)     (1943)
Connection between chromaticnumber  and graph minors.

(2016-05-30)  
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).

(2017-07-28)  
The algebraic  rank of any elliptic curve is equal to its analytic  rank.

This is one of the  7 Millenium Problems  on which the Clay Mathematical Institute has placed a bounty of one million dollars.

(2017-11-13)  
It's true of "sufficiently random" sequences.  Is it true ofprimes ?

François Proth (in 1878) andNormal L. Gilbreath(in 1958) independently considered that a sequence can be obtained from another as the absolutevalue of the differences between consecutive terms. When we start with the sequence of the prime numbers and apply thatprocess iteratively, we obtain the following intriguing table:

This far, all the successive sequences so tabulated start with a leftmost 1. The Proth-Gilbreath conjecture is the unproved statement that it's always so.

In 1878, Proth gave an erroneous proof.

(2019-05-11)  
Proved by Karim Adiprasito  (1988-)  in December 2018.

(2020-06-14)  

In the 1970s, Bill Thurston (1946-2012)  and John Mather (1942-2017) proved a highly nontrivial result: 

In the group  Diff ^r(M)  of the C^r-diffeomorphisms of a compact manifold M, the connected component of the identity is a simple subgroup, in most cases...  The result need not hold when  r  is equal to  dim(M)+1.

In particular,  the case of  Diff ²( S¹)  remainscompletely open.

Do the diffeomorphisms of class  C²  over the circle form a simple group?

The related group denoted  Diff+^1+bv( S¹) is not simple,  as pointed out by the late John Mather. This one is a well-studied group consisting of the orientation-preserving diffeomorphisms f  of the circle, which are  C1 (i.e., the first derivative f '  is a continuous function) with the added condion that  Log f '  is of bounded variation (French:à variation bornée). That group is fundamental indynamical systems,as it meets the premises of Denjoy's theorem,  which wasestablished in 1932 by Arnaud Denjoy (1884-1974),the thesis advisor of the bourbakist Gustave Choquet (1915-2006).

:  Let  G   =   Diff₊^1+bv( S¹)

(2019-12-18)  
Color the plane so two points one unit apart are never the same color.

The problem was made popular by Martin Gardner  in 1960. Since 2018,  we know that the least number of colors needed is  5,  6  or 7.