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(newest | oldest) View (newer 50 | older 50) (20 | 50 | 100 | 250 | 500)- 11:48, 13 May 2022 RandomUndergrad talk contribs created page Right Order Topology on Real Numbers is Topology (Created page with "== Theorem == Let $\tau$ be the '''right order topology on $\R$'''. Then $\tau$ forms a topology on $\R$. That is: :$T = \struct {\R, \tau}$ is a topological space. == Proof == Write $\O = \openint \infty \infty$ and $\R = \openint {-\infty} \infty$. Then $\tau$ can be written as $\set {\openint j \infty: j \in \overline \R}$. First we note that: :$m \le...")
- 08:26, 13 May 2022 RandomUndergrad talk contribs created page Closed Sets of Right Order Space on Real Numbers (Created page with "== Theorem == Let $T = \struct {\R, \tau}$ be the right order space on $\R$. Then $H \subseteq S$ is closed in $T$ {{iff}}: :$H = \O$ or $\R$ or :$H = \hointl {-\infty} a$ for some $a \in \R$. == Proof == By definition of the right order space on $\R$, $U \subseteq S$ is open in $T$ {{iff}}:...")
- 08:26, 13 May 2022 RandomUndergrad talk contribs created page Definition:Right Order Topology on Real Numbers (Created page with "== Definition == <onlyinclude> Let $\tau$ be the subset of the power set $\powerset {\R}$ be defined as: :$\tau := \O \cup \set {\openint a \infty: a \in \R} \cup \R$ Then $\tau$ is the '''right order topology on $\R$'''. Hence the topological space $T = \struct {\R, \tau}$ can be referred to as the '''right order space on $\R$'''. </onlyinclude> == Also see == * Right Order Topology on...")
- 19:23, 12 May 2022 RandomUndergrad talk contribs created page Trace of Product of Matrices (Created page with "== Theorem == Let $\struct {R, +, \circ}$ be a commutative ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$. Let $\mathbf B = \sqbrk b_{n m}$ be an $n \times m$ matrix over $R$. Then: :$\map \tr {\mathbf A \mathbf B} = \map \tr {\mathbf B \mathbf A}$ where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$. == Proof == Let $\math...")
- 09:30, 9 May 2022 RandomUndergrad talk contribs created page Length of Median of Triangle/Proof 3 (Created page with "== Theorem == {{:Length of Median of Triangle}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = {m_c}^2 | r = b^2 + \paren {\frac c 2}^2 - 2 b \paren {\frac c 2} \cos A | c = Law of Cosines }} {{eqn | r = b^2 + \frac {c^2} 4 - 2 b \paren {\frac c 2} \paren {\frac {b^2 + c^2 - a^2} {2 b c} } | c = Law of Cosines }} {{eqn | r = b^2 + \frac {c^2} 4 - \frac {b^2 + c^2 - a^2} 2 | c = }} {{eqn | r = \frac {a^2} 2 + \frac {b^2} 2 - \fr...")
- 19:31, 8 May 2022 RandomUndergrad talk contribs created page Skewness of Binomial Distribution (Created page with "== Theorem == <onlyinclude> Let $X$ be a discrete random variable with a Binomial distribution with parameter $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$. Then the skewness $\gamma_1$ of $X$ is given by: :$\gamma_1 = \dfrac {1 - 2 p} {\sqrt {n p q} }$ where $q = 1 - p$. </onlyinclude> == Proof == From Skewness in terms of Non-Central Moments: :$\gamma...")
- 08:25, 10 April 2022 RandomUndergrad talk contribs created page Talk:Henry Ernest Dudeney/Puzzles and Curious Problems/319 - The Ten Cards/Solution (Created page with "$000X000$, $00X00X0X0$ and $0X00X000$ are shown to be winning positions for $A$. It can be checked that the cases where the second/fourth card is flipped over can also be reduced to these three cases in turn 3. Moreover, $A$ loses if the first/fifth card is flipped, because $B$ can force a position of: :$X0000X0000$ and whatever $A$ does, $B$ can transform it into one of the winning positions above (or more simply, $B$ can copy what $A$ does for the rest of the game i...")
- 18:55, 4 April 2022 RandomUndergrad talk contribs created page Subset Product/Examples/Subset Product with Empty Set (Created page with "== Example of Subset Product == <onlyinclude> Let $\struct {S, \circ}$ be an algebraic structure. Let $A, B \in \powerset S$. If $A = \O$ or $B = \O$, then $A \circ B = \O$. </onlyinclude> == Proof == We show the contrapositive: Suppose that $A \circ B \ne \O$. Then: :$\exists x: x \in A \circ B$ From definition of Definition:Subset Produc...")
- 05:28, 2 April 2022 RandomUndergrad talk contribs created page Coin Problem (Redirected page to Largest Number not Expressible as Sum of Multiples of Coprime Integers) Tag: New redirect
- 11:17, 30 March 2022 RandomUndergrad talk contribs created page Talk:Smallest Differences between Fractional Parts of Square and Cube Roots (Created page with "[https://tio.run/##HcVBCoAwDADBtygoCaRo9VjzExFEiORSS1FoEd9exT3MhjWq2UMpma2TI4LyQLb/ozuxxHUDXcB2A5r/[email protected] The code in Pari/GP]: y=1;for(i=2,1000000,{x=frac(i^(1/2)-i^(1/3));if(x<y&&x!=0,printf("%u\n%f\n",i,x);y=x);}) gives the output: 2 0.15429251247822188403447811693146972800 10 0.0078429701364956102395999779133680384639 42 0.0047140535214104442261022170834592237132 800 0.0010935802363431912136217823550684183680 1570 0.000...")
- 08:12, 30 March 2022 RandomUndergrad talk contribs created page Category:Replicative Function of x minus Floor of x is Replicative (Created page with "Category:Replicative Functions")
- 08:11, 30 March 2022 RandomUndergrad talk contribs created page Replicative Function of x minus Floor of x is Replicative/Lemma (Created page with "== Theorem == <onlyinclude> Let $x \in \R$. Suppose $x - \floor x < \dfrac 1 n$. Then: :$\floor {x + \dfrac k n} = \dfrac {\floor {n x} } n$ for any $0 \le k \le n - 1$. </onlyinclude> == Proof == We have $n x < n \floor x + 1$. By Number less than Integer iff Floor less than Integer: :$\floor {n x} < n \floor x + 1$ Thus $\floor {n x} \le n \floor x$. From definition of floor function: :$n x \ge n \floor x$ By Number not less...")
- 03:47, 30 March 2022 RandomUndergrad talk contribs created page Category:Open Subset of Locally Connected Space is Locally Connected (Created page with "Category:Topology Category:Locally Connected Spaces")
- 13:16, 29 March 2022 RandomUndergrad talk contribs created page Category:Vitali Theorem (Created page with "Category:Measure Theory")
- 13:14, 29 March 2022 RandomUndergrad talk contribs created page Vitali Theorem/Lemma (Created page with "== Theorem == <onlyinclude> For all real numbers in the closed unit interval $\mathbb I = \closedint 0 1$, define the relation $\sim$ such that: :$\forall x, y \in \mathbb I: x \sim y \iff x - y \in \Q$ where $\Q$ is the set of rational numbers. That is, $x \sim y$ {{iff}} their difference is Definition:Rational Number|r...")
- 12:17, 25 March 2022 RandomUndergrad talk contribs created page Category:Integer as Sums and Differences of Consecutive Squares (Created page with "Category:Squares")
- 12:15, 25 March 2022 RandomUndergrad talk contribs created page Integer as Sums and Differences of Consecutive Squares/Examples (Created page with "== Examples of use of Integer as Sums and Differences of Consecutive Squares == Since the proof above is constructive, we can follow the proof and derive: {{begin-eqn}} {{eqn | l = 15 | r = 3 + 4 + 4 + 4 }} {{eqn | r = \paren {- 1^2 + 2^2} + \paren {3^2 - 4^2 - 5^2 + 6^2} + \paren {7^2 - 8^2 - 9^2 + 10^2} + \paren {11^2 - 12^2 - 13^2 + 14^2} }} {{end-eqn}} and its associated family of solutions: {{begin-eqn}} {{eqn | l = 15 | r = - 1^2 + 2^2 + 3^2 - 4^2...")
- 12:15, 25 March 2022 RandomUndergrad talk contribs created page Integer as Sums and Differences of Consecutive Squares/Historical Note (Created page with "== Historical Note == {{AuthorRef|Wacław Sierpiński}} attributes this result to {{AuthorRef|Paul Erdős}} and an M. Surányi. Category:Historical Notes Category:Integer as Sums and Differences of Consecutive Squares")
- 06:37, 25 March 2022 RandomUndergrad talk contribs created page Integer as Sums and Differences of Consecutive Squares (Created page with "== Theorem == Every integer $k$ can be represented in infinitely many ways in the form: :$k = \pm 1^2 \pm 2^2 \pm 3^3 \pm \dots \pm m^2$ for some (strictly) positive integer $m$ and some choice of signs $+$ or $-$. == Proof == First we notice that: {{begin-eqn}} {{eqn | o = | r = \paren {m + 1}^2 - \paren {m + 2}^2 - \paren {m + 3}^2 + \paren {m + 4}^2 }} {{eqn | r =...")
- 13:38, 23 March 2022 RandomUndergrad talk contribs created page Talk:Smallest Number which is Multiplied by 99 by Appending 1 to Each End (Created page with "My bad for not actually checking the calculation when writing the proof for this: The number in the article of Hunsucker and Pomerance is correct. (With a 5 instead of a 3 in the 5th last digit) This is possibly unrelated, but is the prime factorization really necessary for a multiplication result? --~~~~")
- 13:43, 21 March 2022 RandomUndergrad talk contribs created page Category:Power of 2^10 Minus Power of 10^3 is Divisible by 24 (Created page with "Category:Modulo Arithmetic")
- 13:42, 21 March 2022 RandomUndergrad talk contribs created page Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 1 (Created page with "{{MissingLinks|category}} == Theorem == {{:Power of 2^10 Minus Power of 10^3 is Divisible by 24}} == Proof == {{begin-eqn}} {{eqn | l = 2^{10 n} | r = \paren {2^{10} }^n | c = Power of Power }} {{eqn | r = 1024^n | c = as $2^{10} = 1024$ }} {{eqn | r = \paren {1000 + 24}^n | c = rewriting $1024$ as the sum of a power of $10$ and some integer }} {{eqn | r = \sum_{k \mathop = 0}^n 1000^{n - k} \, 24^k | c = Binomial Theorem }} {{eqn...")
- 13:42, 21 March 2022 RandomUndergrad talk contribs created page Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 2 (Created page with "== Theorem == {{:Power of 2^10 Minus Power of 10^3 is Divisible by 24}} == Proof == For $n = 0$ both powers are $1$, and $1 - 1 = 0$ is divisible by $24$. For $n > 1$: {{begin-eqn}} {{eqn | l = 2^{10 n} - 10^{3 n} | r = 2^{3 n} \paren {2^{7 n} - 5^{3 n} } }} {{eqn | o = \equiv | r = 0 | rr= \pmod 8 | c = because $2^3 \divides 2^{3 n}$ }} {{eqn | l = 2^{10 n} - 10^{3 n} | o = \...")
- 13:42, 21 March 2022 RandomUndergrad talk contribs created page Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 3 (Created page with "== Theorem == {{:Power of 2^10 Minus Power of 10^3 is Divisible by 24}} == Proof == {{begin-eqn}} {{eqn | l = 2^{10 n} - 10^{3 n} | r = \paren {2^{10} }^n - \paren {10^3}^n | c = Power of Power }} {{eqn | r = \paren {2^{10} - 10^3} \sum_{j \mathop = 0}^{n - 1} \paren {2^{10} }^{n - j - 1} \paren {10^3}^j | c = Difference of Two Powers }} {{eqn | r = 24 k | c = where $\ds k = \sum_{j \mathop = 0}^{n - 1} {2^{10} }^{n - j - 1} \paren {10^3...")
- 13:42, 21 March 2022 RandomUndergrad talk contribs created page Power of 2^10 Minus Power of 10^3 is Divisible by 24/Example (Created page with "== Example of Power of $2^{10}$ Minus Power of $10^3$ is Divisible by $24$ == Let us examine arguably the flagship example, the smallest number whose compliance is not obvious nor trivial: :$1 \, 048 \, 576$ This is equal to $2^{20}$, which is equal to $2^{10 \times 2}$, thus one of the valid powers of $2$. We then subtract $10^{3 \times 2}$, or $10^6$: :$1 \, 048 \, 576 - 1 \, 000 \, 000 = 48 \, 576$ and we s...")
- 11:22, 18 March 2022 RandomUndergrad talk contribs created page Divisibility of Common Difference of Arithmetic Sequence of Primes (Created page with "== Theorem == If $n$ terms of an arithmetic sequence are primes, then the common difference must be divisible by all primes less than $n$. == Proof == {{WLOG}} suppose the arithmetic sequence is increasing. We also disregard the trivial case...")
- 09:59, 18 March 2022 RandomUndergrad talk contribs created page Existence of Niven Number for Any Sum of Digits (Created page with "== Theorem == Let $b, s$ be integers such that $b > 1$ and $s > 0$. Then there exists a Niven Number in base $b$ with sum of digits $s$. == Proof == Consider the prime factorization of $b$: :$b = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$ where $a_1, a_2, \dots, a_k \ge 1$. Write: :$s = p_1^{c_1} p_2^{c_2} \...")
- 05:52, 18 March 2022 RandomUndergrad talk contribs created page Exist Term in Arithmetic Sequence Divisible by Number (Created page with "== Theorem == Let $\sequence {a_k}$ be an $n$-term arithmetic sequence in $\Z$ defined by: :$a_k = a_0 + k d$ for $k = 0, 1, 2, \dots, n - 1$ Let $b$ be a (strictly) positive integer such that $b$ and $d$ are coprime and $b \le n$. Then there exists a term in $\sequence {a_k}$ that is divisible by $b$. == Proof == We claim th...")
- 13:07, 17 March 2022 RandomUndergrad talk contribs created page Number times Recurring Part of Reciprocal gives 9-Repdigit/Generalization (Created page with "== Theorem == Let a (strictly) positive integer $n$ be such that the decimal expansion of its reciprocal has a recurring part of period $d$ and no non-recurring part. Let $m$ be the integer formed from the $d$ Definitio...")
- 20:15, 16 March 2022 RandomUndergrad talk contribs created page Category:Conjunction implies Disjunction (Created page with "Category:Conjunction Category:Disjunction")
- 20:14, 16 March 2022 RandomUndergrad talk contribs created page Conjunction implies Disjunction/Proof by Truth Table (Created page with "== Theorem == {{:Conjunction implies Disjunction}} == Proof == <onlyinclude> We apply the Method of Truth Tables. As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations. :$\begin{array}{|ccc|c|ccc|} \hline (p & \land & q) & \implies & (p & \lor & q) \\ \hline \F & \F & \F...")
- 20:14, 16 March 2022 RandomUndergrad talk contribs created page Conjunction implies Disjunction/Proof 2 (Created page with "== Theorem == {{:Conjunction implies Disjunction}} == Proof == <onlyinclude> {{BeginTableau|\paren {p \land q} \implies \paren {p \lor q} }} {{Assumption|1|p \land q}} {{Simplification|2|1|p|1|1}} {{Addition|3|1|p \lor q|2|1}} {{Implication|4||\paren {p \land q} \implies \paren {p \lor q}|1|3}} {{EndTableau}} {{qed}} </onlyinclude> == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Hypothetical Syllogism/Formulation 3/Proof by Truth T...")
- 12:09, 16 March 2022 RandomUndergrad talk contribs created page Equivalent Characterizations of Invertible Matrix (Created page with "== Theorem == Let $\mathbf A$ be a square matrix of order $n$ over a field $K$. {{TFAE}} :$(1):\quad$ $\mathbf A$ is invertible :$(2):\quad$ The transpose of $\mathbf A$ is invertible :$(3):\quad$ $\mathbf A$ row-reduces to t...")
- 18:32, 14 March 2022 RandomUndergrad talk contribs created page Sum of Reciprocals of Sequence of Pairs of Odd Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean (Created page with "== Theorem == {{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} } | r = \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 5} + \dfrac 1 {5 \times 13} + \dfrac 1 {13 \times 34} + \cdots | c = }} {{eqn | r = \phi^{-1} | c = }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\phi$ denotes the golden mean. == Proof == {{begin-eqn}} {{eqn | l = \sum_{k...")
- 12:42, 14 March 2022 RandomUndergrad talk contribs created page Even Integers not Sum of Two Abundant Numbers (Created page with "== Theorem == The even integers which are not the sum of $2$ abundant numbers are: :All even integers less than $24$; :$26, 28, 34, 46$ == Proof == From Sequence of Abundant Numbers, the first few abundant numbers are: :$12, 18, 20, 24, 30, 36, 40, 42, 48$ Immediately we see that...")
- 02:09, 13 March 2022 RandomUndergrad talk contribs created page Category:Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient (Created page with "Category:Fibonacci Numbers Category:Binomial Coefficients")
- 02:07, 13 March 2022 RandomUndergrad talk contribs created page Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient/Proof 1 (Created page with "== Theorem == {{:Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient}} == Proof == <onlyinclude> The proof proceeds by induction. For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition: :$F_{2 n} = \ds \sum_{k \mathop = 1}^n \dbinom n k F_k$ $\map P 0$ is the case: {{begin-eqn}} {{eqn | l = F_0 | r = 0 | c = }} {{eqn | r = \sum_{k \mathop = 1}^0 \dbinom 0 k F_...")
- 02:07, 13 March 2022 RandomUndergrad talk contribs created page Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient/Proof 2 (Created page with "== Theorem == {{:Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \dbinom n k F_k | r = \sum_{k \mathop = 1}^n \dbinom n k \paren {\frac {\phi^k - \hat \phi^k} {\sqrt 5} } | c = Euler-Binet Formula }} {{eqn | r = \frac 1 {\sqrt 5} \paren {\sum_{k \mathop = 1}^n \dbinom n k \phi^k - \sum_{k \mathop = 1}^n \dbinom n k \hat \phi^k} | c = Summation is...")
- 13:41, 12 March 2022 RandomUndergrad talk contribs created page Talk:Henry Ernest Dudeney/Puzzles and Curious Problems/108 - The Nine Digits/Solution (More solutions: new section)
- 13:39, 11 March 2022 RandomUndergrad talk contribs created page Talk:Condition for Concurrency of Three Straight Lines (Linear Algebra Results: new section)
- 19:01, 9 March 2022 RandomUndergrad talk contribs created page Category:Integers whose Number of Representations as Sum of Two Primes is Maximum (Created page with "Category:Prime Numbers")
- 18:58, 9 March 2022 RandomUndergrad talk contribs created page Integers whose Number of Representations as Sum of Two Primes is Maximum/Sufficient Condition (Created page with "== Theorem == <onlyinclude> Let $n$ be a (strictly) positive integer. Suppose we have for some positive integer $k$: :$p_k \# \divides n$ :$n \le 2 {p_{k + 1} }^2$ where $p_k \#$ denotes the product of the first $k$ primes. Then $n$ can be represented as the sum of two primes in the maximum number of ways....")
- 12:46, 9 March 2022 RandomUndergrad talk contribs created page Talk:Restriction of Associative Operation is Associative ("Revert": new section)
- 02:46, 9 March 2022 RandomUndergrad talk contribs created page Talk:Number of Digits in Number (Created page with "Should this page and Number of Digits to Represent Integer in Given Number Base be merged together as two equivalent formulations of the same result? --~~~~")
- 12:33, 8 March 2022 RandomUndergrad talk contribs created page Four Fours/150/Solutions/2 (Created page with "== Puzzle: Four Fours: $150$ == {{:Four Fours/150}} == Solution == <onlyinclude> :$150 = \dfrac {\paren {\sqrt 4 \times \sqrt 4}!} {.4 \times .4}$ </onlyinclude> Category:Four Fours/150")
- 13:03, 7 March 2022 RandomUndergrad talk contribs created page Category:Lamé's Theorem (Created blank page)
- 13:03, 7 March 2022 RandomUndergrad talk contribs created page Lamé's Theorem/Lemma (Created page with "== Theorem == Let $a, b \in \Z_{>0}$ be (strictly) positive integers. Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$. Suppose it takes $n$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$. Then $\min \set {a, b} \ge F_{n + 1}$, where $F_n$ denotes the $n$-th Fibonacci number. == Proof == {{WLOG}} suppose $a > b$....")
- 07:24, 7 March 2022 RandomUndergrad talk contribs created page Four Fours/198/Solutions/2 (Created page with "== Puzzle: Four Fours: $198$ == {{:Four Fours/198}} == Solution == <onlyinclude> :$198 = \dfrac {44 \sqrt 4} {. \dot 4}$ </onlyinclude> Category:Four Fours/198")
- 05:14, 7 March 2022 RandomUndergrad talk contribs created page Talk:Construction of Smith Number from Prime Repunit (Created page with "$3304$ is not the smallest multiplier. According to https://primes.utm.edu/glossary/page.php?sort=SmithNumber, we have a list of multipliers, smallest of which is $1540$. The webpage has made no claims on whether $1540$ is the smallest working multiplier. (Curiously the webpage does not list $3304$ even though it works.) $3304$ could be the smallest multiplier that gives the digit sum $n + 27$? It shouldn't be hard to crunch through a few thousand numbers, so I will...")
- 04:47, 7 March 2022 RandomUndergrad talk contribs created page Category:Smith Numbers are Infinite in Number (Created blank page)