For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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8
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116 views

Intuition behind Erdős proof of the infinitude of prime numbers

Suppose by contradiction that there are finitely many primes, namely $p_1, p_2,...,p_k$, where $k$ is a natural number. Now consider another natural number $n$, and all natural numbers $m \leq n$. ...
7
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0answers
75 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
7
votes
0answers
269 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
6
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0answers
42 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
6
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0answers
57 views

Examples of useful, insightful and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
6
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0answers
110 views

Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be ...
6
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308 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
6
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393 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
5
votes
0answers
87 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
5
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188 views

How to show that this function respects the strict ordering of its input.

Suppose you have a vector $\pmb x=\{x_i\}_{i=1}^n$ where each entry is drawn from a continuous distribution and $n$ is even. Then, denote $i^*=\{1\leq j\neq i\leq ...
5
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159 views

Are these sets with orbits subgroups of $S_A$ = set of all permutations of A? - Fraleigh p. 86 8.41-8.43

In Exercises 40 through 43, let $A$ be a set, $B$ a subset of $A$, and let $b$ be one particular element of $B$. Determine whether the given set is sure to be a subgroup of $S_A$ under the induced ...
5
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207 views

Questions on Answer to “The cardinality of the set of all finite subsets of an infinite set”

Would someone please enlarge on Arturo Magidin's original answer ? $1.$ Say the question didn't divulge $|S| = |X|$. Then how can $|S|$ be determined? Any intuition? I recast it below with more ...
5
votes
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141 views

Cauchy induction: are there examples of cases where choosing an integer other than $2$ is a better strategy?

Cauchy induction, sometimes called backwards induction, works as follows: show that $p(1)$ is true show that $p(n)$ implies $p(2n)$ (which inductively implies $p(2^n)$ is true) show that $p(n)$ ...
4
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50 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...
4
votes
0answers
50 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv ...
4
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0answers
76 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
4
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0answers
113 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
4
votes
0answers
75 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
4
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53 views

Nontrivial homomorphism for $Z_a \times Z_b $to $Z_c \times Z_d$ - Fraleigh p. 134 13.35

This isn't a duplicate of this. Let $(A, B) \in \mathbb{Z_a \times Z_b}$. Hinging on p. 2, I guess homomorphism is $h(A,B) = (A \text{ mod } c, B \text{ mod } d)$. I'm unsettled. p. 2 sprang it up ...
4
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0answers
48 views

Counterexamples to Nonidentities - Power of Cosets and Right Coset - Fraleigh p. 103 10.30, 33

Let $H \le$ group G and $a, b \in G.$ Prove or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ and $H = \{(1), (1,3)\}$. ...
4
votes
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170 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
4
votes
0answers
55 views

representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
4
votes
0answers
94 views

Infinite “String” of Implication Statements

This question is inspired by the conversations at Does this require transfinite induction? First of all, does an infinite string of implication statements have a conclusion? I don't think so, but I ...
4
votes
0answers
76 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
4
votes
0answers
119 views

Using other fields of math to simplify a proof.

One of the first non-trivial results given in most courses on algebraic topology is the proof of the Fundamental Theorem of Algebra using topological methods. This is on page 11 of J.P. May's A ...
4
votes
0answers
248 views

Combinatorics and graph theory - counting connected graphs

We denote by $C(n,n+k)$ the number of connected graphs on $n$ vertices with $n+k$ edges. I have 2 problems I wish to prove, but after much effort have gotten nowhere with. I would greatly value some ...
4
votes
0answers
363 views

Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
4
votes
0answers
370 views

Constructing adjunction from left adjoint and unit

The definition of adjoint functors in terms of universal morphisms lends itself to very economical proofs in situations where one has a functor but no "direct" candidate for the left adjoint functor ...
3
votes
0answers
25 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
3
votes
0answers
37 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that ...
3
votes
0answers
132 views

Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless networks specifically when calculating coverage probabilities - (please check page 5 equation 2.11 of the document whose ...
3
votes
0answers
40 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
3
votes
0answers
27 views

Validity of this geometry proof

In triangle ABC, D is the midpoint of AB and E is the point of trisection of BC nearer to C. Given that $\angle$ADC = $\angle$ BAE find $\angle$BAC. Source Q5 Lets call the ...
3
votes
0answers
63 views

Theorem cannot be proven directly

Can we ever prove a theorem cannot be proven directly (i.e. We must use contrapositive or prove by contradiction.)? Can we even rigorously defined whether a proof is direct or not? Example: I was ...
3
votes
0answers
52 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
3
votes
0answers
190 views

Which is the best transitional mathematics book for self-teaching among the ones listed?

What is Mathematics, An Elementary Approach to Ideas and Methods - Courant Robbins Stewart How to Solve It, A New Aspect of Mathematical Solving - Polya Introductory Mathematics, Algebra and Analysis ...
3
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0answers
54 views

Verifying the Cofinite topology

I'm trying to provee that the cofinite topology is a valid topological space. I've defined it as $$C=\{\emptyset\}\cup\{S\subseteq X: X-S \quad\text{is finite }\}.$$ Now, clearly $\emptyset$ and ...
3
votes
0answers
70 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
3
votes
0answers
91 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
3
votes
0answers
64 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
3
votes
0answers
154 views

Substitute Proof by Contradiction/Negation with Direct Proof or Proof by Contrapositive

When do proof by contradiction, or by negation, or direct proof, or proof by contrapositive all work equally well? When can any one be chosen as the proof form? I'm interested in a pragmatic answer ...
3
votes
0answers
50 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
3
votes
0answers
94 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
3
votes
0answers
71 views

Induction on Primitive Recursive Function

The set $F_{n}$ of primitive recursive function symbols of arity $n$ can be defined inductively as: \begin{align} & Z, \text{Succ} \in F_{1} & \\ &\pi_{j}^{n} \in F_{n} \quad \text{for ...
3
votes
0answers
54 views

Verifying an integral identity related to asymptotic homogenization of an elliptic partial differential equation

Background I'm reading Hornung (1997)'s Homogenization and porous media, pg 3: We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, ...
3
votes
0answers
62 views

How does Hildebrands proof of the prime number theorem via large sieve work?

How does the sieve inequality (I may not know the most general form) lead to the distribution of primes? To me, these concepts do not seem to be related. Can their connection be described in a ...
3
votes
0answers
66 views

Is there a proof reading website where users can submit their problems and upload their proof documents to be reviewed?

I'm just looking for a proof reading website that would allow a user to post her/his problem to be proved and her/his scanned documents to their claimed proof to the problem. I was just about to post ...
3
votes
0answers
86 views

has any cycle found in MD5?

We are not sure whether MD5 has fixed point or not. But since the sample space is finite, it must have cycles: $$ A →(MD5)→ B →(MD5)→ C →(MD5)→ D →(MD5)→ A $$ Has any research been done on MD5 to ...
3
votes
0answers
358 views

Confusing proof of brun's theorem?

I read Brun's proof of Brun's theorem here : http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f110.image (and the following pages) and here http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f138.image ...
3
votes
0answers
213 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...