# 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 27,854 total.

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### 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 Power of 2
- 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
- 1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm
- 1-Sequence Space is Proper Subset of 2-Sequence Space
- 1-Sequence Space is Separable
- 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
- 101 is Smallest Number whose Period of Reciprocal is 4
- 103 is Smallest Prime whose Period of Reciprocal is One Third of Maximal
- 1089 Trick
- 11 is Only Palindromic Prime with Even Number of Digits
- 1105 as Sum of Two Squares
- 12 Knights to Attack or Occupy All Squares on Chessboard
- 12 times Divisor Sum of 12 equals 14 times Divisor Sum 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

### 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
- 23 is Largest Integer not Sum of Distinct Perfect Powers
- 239 is not Expressible as Sum of Fewer than 19 Fourth Powers
- 24 divides a(a^2 - 1) when a is Odd
- 24 divides Square of Odd Integer Not Divisible by 3 plus 23
- 24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 1
- 24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 2
- 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
- 27 is Smallest Number whose Period of Reciprocal is 3

### 3

- 3 is Divisor of one of n, n+2, n+4
- 3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers
- 3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal
- 3-Digit Numbers forming Longest Reverse-and-Add Sequence
- 3-Digit Permutable Primes
- 31 is Smallest Prime whose Reciprocal has Odd Period
- 319 is not Expressible as Sum of Fewer than 19 Fourth Powers
- 333,667 is Unique Period Prime with Period 9
- 3367 Multiplied by 2-Digit Number
- 360 divides a^2 (a^2 - 1) (a^2 - 4)
- 37 is Second Number whose Period of Reciprocal is 3
- 399 is not Expressible as Sum of Fewer than 19 Fourth Powers
- 3^x + 4^y equals 5^z has Unique Solution

### 4

- 4 Consecutive Integers cannot be Square-Free
- 4 Integers whose Euler Phi Value is 10,368
- 4 Pints from 5 Pints and 3 Pints
- 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
- 4 Sine Pi over 10 by Cosine Pi over 5/Proof 3
- 40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7
- 479 is not Expressible as Sum of Fewer than 19 Fourth Powers

### 5

- 5 Numbers such that Sum of any 3 is Square
- 5040 is Product of Consecutive Numbers in Two Ways
- 510,510 is Product of 4 Consecutive Fibonacci Numbers
- 559 is not Expressible as Sum of Fewer than 19 Fourth Powers
- 5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3
- 5th Cyclotomic Ring is not a Unique Factorization Domain

### 9

### A

- A.E. Equal Positive Measurable Functions have Equal Integrals
- A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 1
- A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 2
- A.E. Equal Positive Measurable Functions have Equal Integrals/Proof 1
- A.E. Equal Positive Measurable Functions have Equal Integrals/Proof 2
- 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 Limit Theorem
- Abel's Limit Theorem/Proof 1
- Abel's Test for Uniform Convergence
- Abel-Ruffini 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 Order Twice Odd has Exactly One Order 2 Element/Proof 1
- Abelian Group of Order Twice Odd has Exactly One Order 2 Element/Proof 2
- 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
- Abelianization of Free Group is Free Abelian Group
- Abelianization of Group is Abelian
- Abi-Khuzam Inequality
- Abnormal Subgroup is Self-Normalizing Subgroup
- Abridged Multiplication/Examples/Arbitrary Example
- Absolute Continuity of Complex Measure in terms of Jordan Decomposition
- Absolute Continuity of Measures is Transitive Relation
- Absolute Continuity of Signed Measure in terms of Jordan Decomposition
- Absolute Difference Function is Primitive Recursive
- Absolute Net Convergence Equivalent to Absolute Convergence
- Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Convergence implies Absolute Net Convergence
- Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence
- 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 Continuous
- 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 Continuous Real Function is Continuous
- 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 Divergent Infinite Product
- Absolute Value of Even Power
- Absolute Value of Function is Composite with Absolute Value Function
- Absolute Value of Integer is not less than Divisors
- Absolute Value of Integer is not less than Divisors/Corollary
- Absolute Value of Integer is Primitive Recursive
- Absolute Value of Martingale is Submartingale
- Absolute Value of Measurable Function is Measurable
- Absolute Value of Negative
- Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1
- Absolute Value of Power
- Absolute Value of Real-Valued Random Variable is Real-Valued Random Variable
- Absolute Value of Signed Measure Bounded Above by Variation
- 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