Summary of Result  Name  Subject 
Used to prove that there are uncountably many irrational numbers.  
In a right angled triangle, the square of the hypotenuse, equals the sum of the squares of the two right angled edges of the triangle.  
A conjecture concerning the distribution of the zeros of the classical zeta function.  
For n > 2, no three integers a, b, c satisfy: a^n + b^n = c^n.  
Every even integer greater than 2 can be expressed as the sum of two primes.  
Any convex polyhedron with V vertices, F faces, and E edges is such that: V  E + F = 2.  
There are infinitely many prime numbers. In particular if there were finitely many, then the number obtained from multiplying all the primes then adding 1, would also be prime.  
Every non self intersecting loop in the plane divides the plane into an inside region and an outside region.  
Every continuous function from a closed ball in Euclidean space, into itself, has a fixed point.  
Any colouring of a sufficiently large complete graph contains a monochromatic complete subgraph of a given size. o  
If parallel lines are drawn at a distance t apart, and a needle of length L < t is dropped, then with probability 2L /(pi)t, the needle will intersect a line.  
The complement of the union of sets A and B, is the same as the intersection of the complements of A and B.  
The number of elements in any subgroup of a finite group, divides the number of elements in the group.  
Every partially ordered set in which every totally ordered subset has an upper bound, contains at least one maximal element.  
Every simply connected closed 3manifold is homeomorphic to the 3sphere.  
 Summary of Result  Name  Subject 
L^p is a complete normed vector space.  
There does not exist a set whose members are exactly those sets that are not members of themselves  
Given an ergodic dynamical system, then the space average equals the time average almost everywhere.  
For sufficiently large n, then the number of distinct prime factors of n has the normal distribution with mean and variance log ( log (n) ).  
The sequence of prime numbers contains arithmetic progressions of arbitrary length.  
If A and B are complete measure spaces, f an integrable function on A x B, then the integral of f with respect to the product measure, is the iterated integral of f on A, then B.  
A subset of euclidean nspace is compact if and only if it is closed and bounded.  
If two paths connect the same two points, and a function is holomorphic everywhere between them, then the two path integrals of the function are equal.  
There are exactly n^(n2) distinct trees on a set of n labeled vertices.  
Given events A and B, and a probability measure P, then: P(A  B) P(B) = P(B  A) P(A).  
(cos(x) + i sin(x) )^n = cos(nx) + i sin(nx)  
Given an integer n, if n is even divide it by 2. If it is odd, multiply by 3 and then add 1. Continuing in this way, you always eventually reach 1.  
No consistent system of axioms whose theorems can be listed by an 'effective procedure' is capable of proving all facts about the natural numbers.  
Given a sequence of independent identically distributed random variables, then a tail event determined by this sequence either surely happens, or surely does not happen.  
Given sets A, B then if A ≤ B, and B ≤ A, then: A = B.  
