# Tagged Questions

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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### 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 $p(n)$ ...
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### 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 $m \leq n$. ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $k$, ...
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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 $i^*=\{1\leq j\neq i\leq ... 0answers 508 views ### Sum of odd prime and odd semiprime as sum of two odd primes? How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes$(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 ...
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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 ...
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### Number of lucky primes

The "lucky numbers" can be constructed with this sieve. The red ones are the lucky numbers. As you can see, some are prime. Is the number of lucky primes infinite? Edit: Apparently this is an open ...
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### Why is this a corollary of this theorem?

Lang - Algebra p.251 Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $Aut(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over ...
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### Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
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### List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
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### 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 ...
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 ...