Summary of Result | Name |
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. If there were finitely many, then the number obtained from multiplying all the primes then adding 1, would be divisible by a new 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 coloring of a sufficiently large complete graph contains a monochromatic complete subgraph of a given size. | |
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 3-manifold is homeomorphic to the 3-sphere. | |
| Summary of Result | Name |
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 n-space 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^(n-2) 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|. | |
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