# Category:Proven Results

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A proof which invokes the `{{qed}}`

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## Pages in category "Proven Results"

The following 200 pages are in this category, out of 20,172 total.

(previous page) (next page)### 0

### 1

- 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways
- 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways/Proof 1
- 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways/Proof 2
- 1 is Limit Point of Sequence in Sierpiński Space
- 1 plus Perfect Power is not Prime Power except for 9
- 1 plus Power of 2 is not Perfect Power except 9
- 1 plus Square is not Perfect Power
- 1+1 = 2
- 1+1 = 2/Proof 1
- 1+1 = 2/Proof 2
- 1+2+...+n+(n-1)+...+1 = n^2
- 1+2+...+n+(n-1)+...+1 = n^2/Proof 1
- 1+2+...+n+(n-1)+...+1 = n^2/Proof 2
- 1+2+...+n+(n-1)+...+1 = n^2/Proof 3
- 1+2+...+n+(n-1)+...+1 = n^2/Proof 4
- 10 Consecutive Integers contain Coprime Integer
- 10 is Only Triangular Number that is Sum of Consecutive Odd Squares
- 100 in Golden Mean Number System is Equivalent to 011
- 100 using Digits from 1 to 9
- 1089 Trick
- 11 is Only Palindromic Prime with Even Number of Digits
- 12 times Sigma of 12 equals 14 times Sigma of 14
- 121 is Square Number in All Bases greater than 2
- 123456789 x 8 + 9 = 987654321
- 123456789 x 9 + 10 = 1111111111
- 132 is Sum of all 2-Digit Numbers formed from its Digits
- 159 is not Expressible as Sum of Fewer than 19 Fourth Powers
- 169 as Sum of up to 155 Squares
- 17 Consecutive Integers each with Common Factor with Product of other 16
- 1782 is 3 Times Sum of all 2-Digit Numbers from its Digits
- User:1is0?

### 2

- 2 to the n is Greater than n Cubed when n is 10 and above
- 2-Digit Numbers divisible by both Product and Sum of Digits
- 2-Digit Numbers forming Longest Reverse-and-Add Sequence
- 2-Digit Permutable Primes
- 2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9
- 2197 is Cube of 13
- 25 as Sum of 4 to 11 Squares
- 2520 equals Sum of 4 Divisors in 6 Ways
- 2601 as Sum of 3 Squares in 12 Different Ways

### 3

### 4

- 4 Consecutive Integers cannot be Square-Free
- 4 Integers whose Euler Phi Value is 10,368
- 4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value
- 4 Sine Pi over 10 by Cosine Pi over 5
- 4 Sine Pi over 10 by Cosine Pi over 5/Proof 1
- 4 Sine Pi over 10 by Cosine Pi over 5/Proof 2
- 40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7

### 5

### A

- A.E. Equal Positive Measurable Functions have Equal Integrals
- User:Abcxyz/Sandbox/Dedekind Completions of Archimedean Ordered Groups
- User:Abcxyz/Sandbox/Dedekind Completions of Ordered Sets
- User:Abcxyz/Sandbox/Real Numbers/Identity for Real Addition
- User:Abcxyz/Sandbox/Real Numbers/Identity for Real Multiplication
- User:Abcxyz/Sandbox/Real Numbers/Inverses for Real Addition
- User:Abcxyz/Sandbox/Real Numbers/Inverses for Real Multiplication
- User:Abcxyz/Sandbox/Real Numbers/Ordering on Real Numbers is Compatible with Addition
- User:Abcxyz/Sandbox/Real Numbers/Ordering on Real Numbers is Total Ordering
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Associative
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Closed
- User:Abcxyz/Sandbox/Real Numbers/Real Addition is Commutative
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication Distributes over Addition
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Associative
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Closed
- User:Abcxyz/Sandbox/Real Numbers/Real Multiplication is Commutative
- User:Abcxyz/Sandbox/Real Numbers/Real Numbers are Dedekind Complete
- Abel's Lemma/Formulation 1
- Abel's Lemma/Formulation 1/Corollary
- Abel's Lemma/Formulation 2
- Abel's Lemma/Formulation 2/Corollary
- Abel's Lemma/Formulation 2/Proof 1
- Abel's Lemma/Formulation 2/Proof 2
- Abel's Theorem
- Abelian Group Factored by Prime
- Abelian Group Factored by Prime/Corollary
- Abelian Group Induces Commutative B-Algebra
- Abelian Group Induces Entropic Structure
- Abelian Group is Simple iff Prime
- Abelian Group of Order Twice Odd has Exactly One Order 2 Element
- Abelian Group of Prime-power Order is Product of Cyclic Groups
- Abelian Group of Prime-power Order is Product of Cyclic Groups/Corollary
- Abelian Group of Semiprime Order is Cyclic
- Abelian Quotient Group
- Abelianization of Free Group is Free Abelian Group
- Abnormal Subgroup is Self-Normalizing Subgroup
- Abridged Multiplication/Examples/Arbitrary Example
- Absolute Difference Function is Primitive Recursive
- Absolute Value Function is Completely Multiplicative
- Absolute Value Function is Completely Multiplicative/Proof 1
- Absolute Value Function is Completely Multiplicative/Proof 2
- Absolute Value Function is Completely Multiplicative/Proof 3
- Absolute Value Function is Completely Multiplicative/Proof 4
- Absolute Value Function is Convex
- Absolute Value Function is Convex/Proof 1
- Absolute Value Function is Convex/Proof 2
- Absolute Value Function is Even Function
- Absolute Value Function on Integers induces Equivalence Relation
- Absolute Value induces Equivalence Compatible with Integer Multiplication
- Absolute Value induces Equivalence not Compatible with Integer Addition
- Absolute Value is Bounded Below by Zero
- Absolute Value is Many-to-One
- Absolute Value is Norm
- Absolute Value of Absolutely Continuous Function is Absolutely Continuous
- Absolute Value of Absolutely Convergent Product is Absolutely Convergent
- Absolute Value of Complex Cross Product is Commutative
- Absolute Value of Complex Cross Product is Commutative/Examples
- Absolute Value of Complex Cross Product is Commutative/Examples/2+5i cross 3-i
- Absolute Value of Complex Dot Product is Commutative
- Absolute Value of Complex Dot Product is Commutative/Examples
- Absolute Value of Complex Dot Product is Commutative/Examples/2+5i dot 3-i
- Absolute Value of Components of Complex Number no greater than Root 2 of Modulus
- Absolute Value of Convergent Infinite Product
- Absolute Value of Cut is Greater Than or Equal To Zero Cut
- Absolute Value of Cut is Zero iff Cut is Zero
- Absolute Value of Definite Integral
- Absolute Value of Divergent Infinite Product
- Absolute Value of Even Power
- Absolute Value of Integer is not less than Divisors
- Absolute Value of Integer is not less than Divisors/Corollary
- Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1
- Absolute Value of Power
- Absolute Value of Product
- Absolute Value of Product/Proof 1
- Absolute Value of Product/Proof 2
- Absolute Value of Simple Function is Simple Function
- Absolute Value of Simple Function is Simple Function/Proof 1
- Absolute Value of Simple Function is Simple Function/Proof 2
- Absolute Value on Ordered Integral Domain is Strictly Positive except when Zero
- Absolutely Continuous Function is Continuous
- Absolutely Continuous Function is Uniformly Continuous
- Absolutely Convergent Complex Series/Examples/(z over (1-z))^n
- Absolutely Convergent Generalized Sum Converges
- Absolutely Convergent Product Does not Diverge to Zero
- Absolutely Convergent Product Does not Diverge to Zero/Proof 1
- Absolutely Convergent Product Does not Diverge to Zero/Proof 2
- Absolutely Convergent Product is Convergent
- Absolutely Convergent Series is Convergent
- Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach
- Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Necessary Condition
- Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Sufficient Condition
- Absolutely Convergent Series is Convergent/Complex Numbers
- Absolutely Convergent Series is Convergent/Real Numbers
- Absorption Laws (Boolean Algebras)
- Absorption Laws (Logic)/Conjunction Absorbs Disjunction
- Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Forward Implication
- Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Proof 1
- Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Proof 2
- Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Reverse Implication
- Absorption Laws (Logic)/Disjunction Absorbs Conjunction
- Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Forward Implication
- Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 1
- Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 2
- Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Reverse Implication
- Absorption Laws (Set Theory)/Corollary
- Absorption Laws (Set Theory)/Intersection with Union
- Absorption Laws (Set Theory)/Intersection with Union/Proof 1
- Absorption Laws (Set Theory)/Intersection with Union/Proof 2
- Absorption Laws (Set Theory)/Union with Intersection
- Absorption Laws (Set Theory)/Union with Intersection/Proof 1
- Absorption Laws (Set Theory)/Union with Intersection/Proof 2
- Abstract Model of Algebraic Extensions
- Abundancy Index of Product is greater than Abundancy Index of Proper Factors
- Abundancy of Integers in form 945 + 630n
- Acceleration Due to Gravity
- Acceleration is Second Derivative of Displacement with respect to Time
- Acceleration of Rocket in Outer Space
- Acceleration Vector in Polar Coordinates
- Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit
- Accumulation Point of Sequence is not necessarily Limit Point
- Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range
- Accumulation Point of Sequence of Reciprocals and Reciprocals + 1
- Accumulation Points for Sequence in Particular Point Space
- Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space
- Accuracy of Convergents of Continued Fraction Expansion of Irrational Number
- Accuracy of Convergents of Continued Fraction Expansion of Irrational Number/Corollary
- Accuracy of Convergents of Convergent Simple Infinite Continued Fraction
- Action of Group on Coset Space is Group Action
- Action of Inverse of Group Element
- Adding Edge to Tree Creates One Cycle
- Addition is Primitive Recursive
- Addition Law of Probability