Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

15776 questions
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How do I rephrase this implication to be less hand-wavy?

I have a homework problem asking me to prove that it's impossible to construct a set of integers so that the set has certain characteristics. I've showed the following things: It's impossible to construct a set of integers modulo 100 so that the…
Rag
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Proof by Induction - Wrong common factor

I'm trying to use mathematical induction to prove that $n^3+5n$ is divisible by $6$ for all $n \in \mathbb{Z}^+$. I can only seem to show that it is divisible by $3$, and not by $6$. This is what I have done: Let $f(n) = n^3+5n$. Basis Step: When…
Anon Anon
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Finding a proof or a counter example in a programming puzzle

Some years ago I entered a programming contest and this was one of the problems: Binary Granny Summary: Given a positive integer N find 2 positive integers such that $$ x + y = N $$Let X and Y be the number of bits set in the base 2 representation…
ICTylor
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Proof by Geometric Intuition may Fail!!

In a book on differential forms, I read, "After all, there were times when people took geometric intuition as proof, and later found that their intuition was wrong". I would like to see an example where intuitive geometric proof fails. (In…
Groups
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Proof involving Induction

Prove that for every integer n ≥ 1, we have $$ \sum_{i=1}^ni^3=\left(\frac{n(n+1)}2\right)^2 $$ Solve using Mathematical Induction, include the Inductive Step Base Case is that both the left and right side $=1$ when $n=1$. and the Inductive…
Drew
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Proving directly that ($a+b)^3 \equiv a^3 + b^3 \mod 3$

Assuming a and b are integers, I must prove directly that: $$ (a + b)^3 \equiv (a^3 + b^3) \mod 3 $$ First, my peers and I made the mistake of assuming what we are trying to prove and thus failed. I've tried expanding $(a + b)^3$ into $a^3 + b^3…
Blue Wizard
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Is what I've done a proof? Proving there is always an rational number between two distinct rational numbers

The exercise I am working on is about proving whether there is always a rational number between two other distinct rational numbers. I came up with this $\frac{a}{b} < \frac{ad + bc}{2bd} < \frac c d$ But is what I've written a proof, or is it just…
Tim H UK
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Proof by induction of a sum?

Let $n ∈ N$. Prove by induction that there are $n$ ways to write the number $n$ as a sum $n=x_1+x_2+...+x_k$ where the $x_i$ are natural numbers and $x_1 ≤x_2 ≤...≤x_k ≤x_1+1$. For example, $5 = 5$, $5 = 1 + 1 + 1 + 1 + 1$, $5 = 1 + 1 + 1 + 2$, $5…
Test
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Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ

I can do it by induction, thanks to the wonderful people of this website, but I'm not sure how to do it by the Division algorithm. Can anyone help me? I think I can show how 3 divides 2n, but I'm not sure how to show that 3 divides n³.
JCMcRae
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Prove that if $p\ge 5$ is prime, then $p^2 + 1$ is composite

So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$? As in, making it $(6q+1)^2+1$ or $(6q+5)^2+1$?
JCMcRae
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Suppose n is an integer. Use a proof by contrapositive to show if n^3 is even, then n is even

So, we assume that n is not even. Then, $n$ is odd, so $n= 2k+1$ for some integer $k$. Then, $(2k+1)^3 = 8x^3+12k^2+6k+1$. Would it be legal, then, for me to say $(8k^3+12k^2+6k)+1 = 2(4k^3+6k^2+3k)+1$, and then say that since $4k^3+6k^2+3k$ is an…
JCMcRae
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Is this correct structural induction (subsets)?

The last question (which is always the hardest) of my Induction Exercises goes like this: Let S be the subset of $\mathbb{Z}$ defined by: -12,20 $\in$ S if x,y $\in$ S, then x+y $\in$ S We use structural induction to show that S = {4k | k $\in…
Tiamo P.
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writing proofs involving matrices

Notation has always been my downfall when it comes to writing proofs. I have been given a general problem to try and write a proof based on this. This is just for research and practice. Given a matrix $e_{ij}A_{nxm}$ If the matrix is multiplied on…
cele
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Why does the solution of a Pell's equation equal zero?

A Pell's equation is given in the following way: $ nx^2 + 1 = y^2 $ According to mathematical rules and the website http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html it can also be written like this: $y^2 - nx^2 = 1$ The equation…
Paul
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Very simple proof help about integers

I was wondering if anybody could help me with proving something very simple: that $9n\ne6$ when n is any integer. It seems extremely intuitive but I don't know how to make it into part of a rigorous proof.
hyperdo
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