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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain a proposed proof/solution.)

For questions concerning a specific proof (or a proof sketch) or a solution to some problem; asking a question with this tag indicates one would like answers to respond broadly as to the following:

  • Verification of the proof/solution;
  • Identifying errors in the proof/solution;
  • Suggestions for improving the proof/solution;
  • Alternative approaches.

Also, consider the related tags and .

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Let $s$ and $g>0$ be given integers. Prove that integers $x$ and $y$ exist satisfying $x+y=s$ and $(x,y)=g$ iff $g|s$.

Problem: Let $s$ and $g>0$ be given integers. Prove that integers $x$ and $y$ exist satisfying $x+y=s$ and $(x,y)=g$ iff $g|s$. My Attempt I have already proved the theorem in the "right " direction. So I shall write the converse: Assuming $g|s$…
Student
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What is the negation of "there are infinitely many integer solutions $(x,y)$ where $x$ is odd"

Is the negation "there are infinitely many integer solutions $(x,y)$ where $x$ is even"?
blablabla
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Is it true that $\int_{0}^{1}\sin(m \pi x)\cos(n \pi x) dx = 0$ only when $n+m $ is even?

I would just like to verify this is correct: $$\int_{0}^{1}\sin(m \pi x)\cos(n \pi x) dx = -\frac{1}{2}\bigg[\frac{(-1)^{n+m}}{(n+m)\pi} + \frac{(-1)^{n-m}}{(n-m)\pi} -\frac{1}{(n+m)\pi} - \frac{1}{(n-m)\pi} \bigg]$$ and so the above integral is…
user197848
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Proof of Uniqueness of Two Lines

I am trying to show a proof of the uniqueness of two equations. If $a,b,c,d\in\mathbb R$ and $ad-bc\neq 0$, then or any $\alpha,\beta \in \mathbb R$ the pair of equations: $$ax + by = \alpha\\cx + dy = \beta$$ have a unique solution where $x = x_0$…
Tsangares
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Biconditional proof - Don't know if I'm wrong

Let $n$ be a positive integer. Prove that $n^3 + 1$ is odd if and only if $n^2 − 1$ is odd. What kind of proofs did you use? I chose direct proof; $$n^3 + 1 = (2k+1)^3 +1 = 2(k^3+k^2+k+1) = 2k~$$ $$n^2 - 1 = (2k+1)^2 -1 = 2(2k^2+2k+1) =…
ayylmaoKing69
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Is Integral of an odd function always even?

Suppose we are given $f(x)$ such that $f(-x)=-f(x)$, then if we consider the integral $$F(x)=\int_{0}^{x} f(t)dt$$ Will $F(x)$ be an even function? There are a couple of examples that strengthen this claim, such as $x^3$ and $\sin(x)$, but I'm not…
Student
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Prove that if $n$ is a perfect square, then $n + 2$ is not a perfect square

Assume towards a contradiction that $n+2$ is a perfect square. If $n=4=2*2$ then $n+2 = 6$ is a perfect square. Contradiction! QED. Is this a valid proof? Can I show a contradiction by finding only one example?
Jason
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Proof clarification involving $f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$

Please see the below proof from the Book of Proof by Richard Hammack. I've broken out the step I have a question on and made the text preceding the equation bold. Suppose $f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ is a polynomial of degree $1$…
ClownInTheMoon
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For all even integers $n$, $(-1)^n = 1$

Suppose that $n$ is an even integer. Then $n = 2r$ for some $r \in\mathbb{Z}$. Hence $(−1)^n =(−1)^{2r} = 1$ because $2r$ is even. Therefore $(−1)^n = 1$ for any even integer $n$ Verification?
SunshineTS
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Check for the validity an 'unusual' proof for an inequality

I recently picked up a BMO 2 question from 2005: Let $a$, $b$, $c$ be positive real numbers. Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2 ≥ \left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ However, I wasn't…
Shuri2060
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Order of $g$ is identical with the minimal $d \in \Bbb N_+$ for which there is a group homomorphism $\Bbb Z / (d) \rightarrow G$

Task Let $G$ be a group and $g \in G$ an element with a finite order. Show that the >order of $g$ is identical with the minimal $d \in \Bbb N_+$ for which there is >a group homomorphism $$\Bbb Z / (d) \rightarrow G,$$ where $g$ is a part of its…
Julian
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Handling undefined case in uniqueness proof (How to Prove It, Velleman; 5.6, 2)

Below is a theorem I was asked to prove from Velleman's book How to Prove It, along with my proof. My question is regarding the last step of the uniqueness portion of the proof where I am required to divide by $(y + 1)$. Since $y$ can be any number,…
user339778
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Doubt in the correctness of the proof by induction of the corollary to the Fundamental Theorem of Algebra

I came across the following proof of the corollary of the Fundamental Algebra Theorem, which I shorten as follows: "Every polynomial $p(z) = a_nz^n + a_{n-1}z^{n-1}+...+a_1z+a_0$ has a factorisation $p(z) =…
Gre
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Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$

Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$ My method: Assume that $r \gt 0$, and $1 - r \gt 0$. Hence $r(1 - r) \gt 0$. So, $1 \le 4r (1 - r)$. Hence, $1 \le 4r - 4r^2$. Hence, $4r^2 - 4r + 1 \ge 0$. Thus, $(2r -…
Matt
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Proof by contrapositive help for if $15n$ is even, $9n$ is even

I wanted to check if this was a valid proof for the following: Let $n \in\mathbb Z$. Prove that if $15n$ is even, then $9n$ is even. What I have is the following: $$\forall n\in\mathbb Z,p\left( n\right) \rightarrow q\left( n\right) $$ $${\sim…
PutsandCalls
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