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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0answers
155 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 $m \leq n$. ...
8
votes
0answers
287 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 ...
7
votes
0answers
182 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
7
votes
0answers
135 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
7
votes
0answers
220 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
0answers
322 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
votes
0answers
117 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
votes
0answers
185 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 $p(n)$ ...
6
votes
0answers
399 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
101 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
votes
0answers
191 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
votes
0answers
391 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 ...
5
votes
0answers
381 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 ...
4
votes
0answers
43 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
4
votes
0answers
46 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
4
votes
0answers
82 views

Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
4
votes
0answers
56 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
52 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
votes
0answers
104 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
4
votes
0answers
84 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
votes
0answers
230 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 $\mathscr{C}^1$ on $(c,d)$. Then ...
4
votes
0answers
82 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
votes
0answers
89 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 ...
4
votes
0answers
188 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
97 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
79 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
124 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
276 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 ...
3
votes
0answers
55 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
3
votes
0answers
34 views

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain.

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain. Proof that $\mathbb Z[\sqrt{3}]$ is a Euclidean Domain Is it possible to solve this question without using $\mathbb{Q}[\sqrt{3}]$ restricted ...
3
votes
0answers
150 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
3
votes
0answers
18 views

for a given $f$, $f$ is measurable iff $f^{-1} (${$-\infty$}$) \in \mathcal{M}$ , $f^{-1} (${$\infty$}$) \in \mathcal{M}$ and f is measurable on $Y$.

Let $f : X \rightarrow \bar{\mathbb{R}}$ and $Y = f^{-1}(\mathbb{R})$ then f is measurable iff $f^{-1} (${$-\infty$}$) \in \mathcal{M}$ , $f^{-1} (${$\infty$}$) \in \mathcal{M}$ and f is measurable on ...
3
votes
0answers
35 views

Induction proof - not sure how to proceed with next step

Define two sequences $A_n, B_n$ as follows: \begin{align*} A_1 &= 1\\ A_2 &= 3\\ A_3 &= 2 \cdot 3+1=7 \\ A_4 &= 2 \cdot 7 + 3 = 17\\ A_5 &= 2 \cdot 17 + 7 = 41\\ A_n &= ...
3
votes
0answers
44 views

Showing an infinite intersection of sets has a supremum

If $(A_n)_{n \in \mathbb{N}}$ are sets where each $A_n$ has a supremum and $\bigcap_{n \in \mathbb{N}}A_n \neq \emptyset$, then $\bigcap_{n \in \mathbb{N}}A_n$ has a supremum. I am about 49% sure ...
3
votes
0answers
30 views

To prove that $f(A)$ is compact in $(Y,e)$.

Let $(X,d)$ and $(Y,e)$ be metric spaces, $A \subset X$ is compact and $\eta$ a fixed number and $f : A \to Y$ a function such that $$e(f(x),f(y)) \leq \eta d(x,y) \ , \ \forall x,y \in A$$ To prove ...
3
votes
0answers
60 views

Help with a trigonometric proof, please?

Hexagon $ABCDEF$ is inscribed in the circle of radius $R$ . $AB=CD=EF=R$. Points $I$, $J$, $K$ are the midpoints of segments $\overline{BC}$, $\overline{DE}$, $\overline{FA}$ respectively. Then ...
3
votes
0answers
48 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
3
votes
0answers
104 views

Prove that in a group iterated commutators with repeated generators is trivial implies that each generator commutes with all its conjugates

Let $G$ be a finitely generated group with generating set $S=\{x_1,\cdots,x_n\}$. Let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Suppose that every iterated commutator with repeated ...
3
votes
0answers
43 views

On the supremum of the union of two bounded sets

Let $A,B$ be bounded subsets of an ordered set $S$. Then $A \cup B$ is bounded and $\sup( A \cup B) = \sup \{ \sup A, \sup B \} $. Attempt to solution: Let $x \in A \cup B$. Then $x \in A $ or $x ...
3
votes
0answers
29 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
49 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
58 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
33 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
65 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
56 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
244 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
votes
0answers
78 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
78 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
127 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
166 views

Stereographic projection from sphere to $\mathbb{R}^2$

This question is from my tutorial problem set: One way to define a system of coordinates for the sphere $S^2$ given by $x^2+y^2+(z-1)^2=1$ is to consider the stereographic projection $\pi:S^2-\{N\} ...