Unsolved Problem 1: (Catalan's Conjecture, now known as Mihăilescu's theorem)
Are 8 and 9 the only consecutive perfect powers? 
An integer is a perfect power if it is of the form m^n where m and n are integers and n>1. 
It is conjectured that 8=2^3 and 9=3^2 are the only consecutive integers that are perfect powers.
The conjecture was finally proved by Preda Mihailescu in a manuscript privately circulated on April 18, 2002.  The proof has now appeared in print (Mihailescu 2004) and is widely accepted as being correct and valid (Daems 2003, Mets�nkyl� 2003), according to Mathworld.

Unsolved Problem 2: (Twin Primes Conjecture)
Are there an infinite number of twin primes? 
A prime number is an integer larger than 1 that has no divisors other than 1 and itself.
Twin primes are two prime numbers that differ by 2. For example, 17 and 19 are twin primes. 

Unsolved Problem 3: (The Rational Box)
Does there exist a rectangular box all of whose edges and diagonals are integers? 
By a rectangular box, we mean a solid with six rectangular faces. This common figure is also known as a rectangular parallelepiped.
The diagonals of a box include the face diagonals and the main diagonals.
A face diagonal joins opposite vertices of a face.
A main diagonal (or space diagonal) joins opposite vertices of the box. 
(See Perfect Cuboid and Euler Brick.)

Unsolved Problem 4: (Equichordal Points, was solved in 1996 by M. Rychlik)
Can a closed curve in the plane have more than one equichordal point? 
The line joining two points on a curve is called a chord.
A point inside a closed convex curve in the plane is called an equichordal point if all chords through that point have the same length. For example, the center of a circle is an equichordal point for that circle. 
It was not known for a long time if there is a closed curve that has two distinct equichordal points.
The problem was first proposed by Fujiwara (1916) and Blaschke et al. (1917).

Unsolved Problem 5: (Goldbach's Conjecture)
Is every even integer larger than 2 the sum of two primes? 
A prime number is an integer larger than 1 whose only positive divisors are 1 and itself. 
For example, the even integer 50 is the sum of the two primes 3 and 47.

Unsolved Problem 6: (Prime Fibonacci Numbers)
Are there infinitely many prime Fibonacci numbers? 
A prime number is an integer larger than 1 whose only positive divisors are 1 and itself.
A Fibonacci number is a member of the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
in which each term is the sum of the previous two terms.

Unsolved Problem 7: (Magic Knight's Tour)
Is there a magic knight's tour on an 8 X 8 chessboard? 
A knight's tour of a chessboard is a sequence of moves by a knight such that each square of the board is visited exactly once.
If the successive squares are numbered from 1 to 64, in order, the tour is called a magic tour if the resulting square is a magic square.
A magic square is a square array of numbers such that each row and column and the two main diagonals sum to the same number (the magic constant). 
Semi-magic knight's tours are known (in which the rows and columns sum to the same number, but the diagonals do not sum to that number).

Unsolved Problem 8: (pi+e)
Is pi+e irrational? 
The number pi is the ratio of a circle's circumference to its diameter.
The number e is the base of natural logarithms and is approximately equal to 2.71828. It is the unique number a such that the derivative of a^x is a^x. 
A rational number is a number that is the ratio of two integers. All other real numbers are said to be irrational. 
It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational.

Unsolved Problem 9: (Tiling the Unit Square)
Will all the 1/k by 1/(k+1) rectangles, for k>0, fit together inside a 1 X 1 square? 
Note that the sum of the areas of all these rectangles is 1.

Unsolved Problem 10: (Egyptian Fractions)
If n is an integer larger than 1, must there be integers x, y, and z, such that 4/n=1/x+1/y+1/z? 
A number of the form 1/x where x is an integer is called an Egyptian fraction.
Thus, we want to know if 4/n is always the sum of three Egyptian fractions, for n>1.

Unsolved Problem 11: (Odd Perfect Numbers)
Are there any odd perfect numbers? 
A perfect number is a positive integer that is equal to the sum of all its positive divisors, other than itself.
For example, 28 is perfect because 28=1+2+4+7+14.

Unsolved Problem 12: (Graceful Trees)
Is every tree graceful? 
A graph is a set of points (called vertices) and a set of lines (called edges) joining these vertices.
A tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges.
A graph is said to be graceful if you can number the n vertices with the integers from 1 to n and then label each edge with the difference between the numbers at the vertices, in such a way that each edge receives a different label. 
For example, a graceful numbering is shown for the following tree with 9 vertices: 

  (5)         (1)---(4)
  /           /
(7)---(3)---(9)---(2)
  \     \
  (6)   (8)

The edge labels are the numbers from 1 to 8.

Unsolved Problem 13: (Rational Distances to the Vertices of a Square)
Is there a point in the plane that is at a rational distance from each of the four corners of a unit square? 
A rational number is a number that is the ratio of two integers.
A unit square is a square of side length 1.

Unsolved Problem 14: (A Series involving 1/n^3)
What is the value of 1/1+1/8+1/27+1/64+1/125+...? 
The nth term is the reciprocal of n^3.  The sum is called Apery's contant after Roger Ap�ry (1916�1994) who in 1978 proved it to be irrational.  It is unknown if it is algebraic or transcendental, or if it is a rational multiple of pi^3.
If 3 is replaced by 2, it is known that the series sums to (pi^2)/6.
If 3 is replaced by 4, it is known that the series sums to (pi^4)/90.

Unsolved Problem 15: (Square Free Mersenne Numbers)
Is every Mersenne number square free? 
A Mersenne number is a number of the form (2^p)-1 where p is a prime.
A prime is an integer larger than 1 whose only positive divisors are 1 and itself.
An integer is said to be square free if it is not divisible by a perfect square, n^2, for n>1. 

Unsolved Problem 16: (A Billiards Problem)
Does every obtuse triangle admit a periodic orbit for the path of a billiard ball? 
We assume that the billiard ball bounces off each side so that the angle of incidence equals the angle of reflection. If it hits a vertex, it rebounds along the reflection of its entry path in the angle bisector of the angle at that vertex. The orbit (or trajectory) is periodic, if after a finite number of reflections, it returns to its starting point.

Unsolved Problem 17: (Lattice Points Covered by a Set)
Is there a set S in the plane such that every set congruent to S contains exactly one lattice point? 
A lattice point is a point with integer coordinates. 

Unsolved Problem 18: (Diophantine Equation of Degree Five)
Are there distinct positive integers, a, b, c, and, d such that a^5+b^5=c^5+d^5? 
It is known that 1^3+12^3=9^3+10^3 and 133^4+134^4=59^4+158^4, but no similar relation is known for fifth powers. Other remarkable identities are 27^5+84^5+110^5+133^5=144^5 and 2682440^4+15365639^4+18796760^4=20615673^4.

Unsolved Problem 19: (Pushing Discs Together, solved in 2000 by K. Bezdek and R. Connelly. See their web page.)
When equal sized discs are pushed closer together, can the area of their union increase? 
By a disc we mean a circle and its interior. The result is known to be true for two discs. By being pushed together, we mean that the distance between each pair of discs is smaller after the pushing. The union of a set of discs is the area covered by all of the discs. The discs are allowed to overlap. (The origin of the problem is Kneser (1955) and Poulsen (1954).  The Kneser�Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls.)

Unsolved Problem 20: (Primes of the Form n^2+1)
Are there infinitely many primes of the form n^2+1? 

Unsolved Problem 21: (Sum of Seven Cubes)
Is every integer larger than 454 the sum of seven or fewer positive cubes?

Unsolved Problem 22: (Triangles with Integer Sides, Medians, and Area)
Is there a triangle with integer sides, medians, and area? 
A median of a triangle is the line segment joining a vertex and the midpoint of the opposite side.

Unsolved Problem 23: (Thirteen Points on a Sphere)
How should you locate 13 cities on a spherical planet so that the minimum distance between any two of them is as large as possible? 

Unsolved Problem 24: (Primes Between Consecutive Squares)
Is there always a prime number between any two consecutive squares? 

Unsolved Problem 25: (The Collatz Conjecture)
Start with any positive integer. Halve it if it is even; triple it and add 1 if it is odd. If you keep repeating this procedure, must you eventually reach the number 1? 
For example, starting with the number 6, we get: 6, 3, 10, 5, 16, 8, 4, 2, 1.

Unsolved Problem 26: (Inscribing a Square in a Curve)
Given a simple closed curve in the plane, can we always find four points on this curve that are the vertices of a square?

Unsolved Problem 27: (Factorial that are one less than a Square)
Are there integers n and x (with n>7) such that n!=x^2-1? 
By n! we mean the product of the integers from 1 to n. It is known that 4!+1=25=5^2, 5!+1=121=11^2, and 7!+1=5041=71^2.

Unsolved Problem 28: (Expressing 3 as the Sum of Three Cubes)
The number 3 can be written as 1^3+1^3+1^3 and also as 4^3+4^3+(-5)^3. Is there any other way of expressing 3 as the sum of three (positive or negative) cubes?

Unsolved Problem 29: (Fitting One Triangle Inside Another)
Let triangles A and B have edge lengths a1, a2, a3, and b1, b2, b3, respectively. What is the necessary and sufficient condition on the variables a1, a2, a3, b1, b2, b3 so that triangle A can fit inside triangle B? 

Unsolved Problem 30: (Sum of Four Cubes)
Is every integer the sum of four cubes? 
Here we allow the cubes to be positive, negative, or zero. For example, 84=0^3+41639611^3+(-41531726)^3+(-8241191)^3.  Eugen Dedu said, "It is not known if 148, for example, is the sum of four cubes."  Then it was pointed out that 148 = 67^3 + 1^3 - 56^3 - 50^3, and that a proof exists that any number equivalent to 0,1,2,3,-3,-2, or -1 (mod 9) has a solution.  (See Alpertron: FCUBES.HTM,  http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100, Sloane: A046041,  and Mathworld: Cubic Number.)  It has been conjectured that all numbers equivalent to plus or minus 0, 1, 2, or 3 (mod 9) can be expressed as the sum of three cubes, which means by adding or subtracting the cube of one, all numbers can be expressed as the sum of four cubes (one of them 1), as in the case of 148, above.

Unsolved Problem 31: (Different Number of Distances)
Is it always possible to have n points in the plane (no 3 on a line; no 4 on a circle), such that for every k (with 0 < k < n), there is a distance determined by these points that occurs exactly k times? 
For example, 4 points determine 6 distances. We want one distance to occur just once, another distance to occur twice, and a third distance to occur three times.
So far, configurations have been found for n=2,3,4,...,8.

Unsolved Problem 32: (Can the Cube of a Sum Equal their Product)
Can you find three nonzero integers x, y, and z, such that (x+y+z)^3=xyz?

Unsolved Problem 33: (Unit Triangles in a Given Area)
Is there a constant, A, such that any set in the plane of area A must contain the vertices of a triangle with area 1? 

Unsolved Problem 34: (Squares with Two Different Decimal Digits)
Are there only finitely many perfect squares with just two different nonzero decimal digits? 
For example, 38^2=1444, 88^2=7744, 109^2=11881, 173^2=29929, 212^2=44944, 235^2=55225, and 3114^2=9696996.

Unsolved Problem 35: (Must one of n points lie on n/3 lines?)
If n points in the plane are not collinear, must there be one of those points that lies on at least n/3 of the lines determined by those points? 

Unsolved Problem 36: (Primes of the form n^n+1)
Is there any value of n other than 1, 2, and 4, such that n^n+1 is a prime? 
(It seemed that this problem was solved, but the author of this proof has reportedly admitted that it contains an error.)

Source of this page: lifc.univ-fcomte.fr/~dedu/math/unsolvedPbs.txt by Eugen Dedu