# Questions tagged [proof]

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### The integer ababab (a,b>0) is always divisible by 7, without remainder

The integer $ababab$ $(a>0,b>0)$ is always divisible by $7$, without rest. I tried to prove this by: ...
• 11.3k
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### Why can't Mathematica confirm this simple identity? [closed]

It is well known that for positive integer $n\geq2$, $$\sum_{k=1}^{n-1} \frac{(-1)^{k-1}(k-1)!^2}{(n^2-1^2)\ldots(n^2-k^2)}=\frac{1}{n^2}-\frac{2(-1)^{n-1}}{n^2 \binom{2n}{n}}.$$ This identity ...
• 763
467 views

### Maximum sum of product of two arrays [closed]

Any math proof for maximum sum of product of two array achieved by maximum times maximum A = {5, 2, 1, 3}; B = {1, 2, 3, 4}; ...
• 39
56 views

### proving closure for vector space

I would like to present to my students, closure under vector addition (same situation for scalar multiplication). I have found that this works fine: ...
1 vote
27 views

### Why does Mathematica not return "True" for Correlation function equality expression?

I wish to check the invariance property of correlation. As described in this wiki section Wiki - Mathematical Properties of Correlation. A quote from the Wiki page I wish to test is below. That is, ...
• 353
131 views

### Proving an integral identity [closed]

Assuming $f$ is continus on $[0,\pi]$, show that $$\int_{0}^{\pi }xf(\sin x)dx=\frac{\pi }{2}\int_{0}^{\pi }f(\sin x)dx$$ I tried to demonstrate it using integration by parts, but I did not succeed. ...
45 views

### What is the proper way to feed this function as input and test if my statement is true?

$f(S)$ = {$n∈N$: $n=m²$ for $m∈S$} $S$ = $1,2,3$ Square each element in S Now, $S= 1, 4, 9$ All elements in $S$ are counting from $1$ to $N$. In this case $N=3$ $f(S)$ = {$n∈N$: $n=m²$ for $m∈S$} ...
• 101
76 views

### Prove Geometry automatically [closed]

If given a geometric problem, can Mathematica solve it automatically? Is there any software that can solve geometry problems automatically?
• 137
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### Trigonometric identity not simplifying

I have a pair of numbers {x,y} (Reals or Integers) for which I want to confirm symbolically that ArcTan[y/x]+ArcTan[x/y]==Pi/2 Here is my attempt: ...
• 1,539
1 vote
71 views

### Can Mathematica check if I correctly applied an algebraic transformation?

Mathematica has tons of functions to perform algebraic manipulations. But is there a way to use it to check if I didn't make mistakes in my own algebraic manipulations. As a real world example: [...
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### How to "Prove" this summation result?

I have this messy function with $n$, $k$, $i$ integers: $$r(\rm n,k)=\frac{k 2^{1-2 \rm{n}} (2 k)! (-2 k+2 \rm{n}+1) (2 \rm{n}-2 k)!}{(k!)^2 \left(1-4 (i-k)^2\right) ((\rm{n}-k)!)^2}$$ I want to ...
• 6,898
110 views

### How to use Resolve to prove ForAll function

I want to prove that ∀n ∈ N，Mod[9^n - 8 n - 1, 64] == 0 I think I should use Resolve to prove it ...
• 169
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### How to extract steps from FullSimplify? [duplicate]

The answer proposed to the original question doesn't solve it, I need to understand the results of TraceInternal when used on FullSimplify as a sequence of steps, which has not been resolved for me. ...
• 30.3k
431 views

### Symbolically prove that two expressions are identical

I encountered this problem when trying to reproduce the result of this paper. (The relevant parts are all included in the preview i.e. the 1st page of the article. This link is just given as ...
• 51.5k
214 views

### Proving that Reverse[Reverse[x]]]=x

It is clear that for any list $x$, Reverse[Reverse[x]]=x. I want to have Mathematica tell me that this is true. I have tried entering ...
• 153
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### How to prove that an equation can be divided by any n?

I'm new to Mathematica and came across with the problem. I can't prove that (for example) 2*n is always even no matter what n is....
197 views

### Theorem Proving

It's a hard for me to write english so I'll try to go straight to my problem. There is abstract predicate f, that takes two parameters. There are also two other ...
• 151
223 views

### Fundamental question about capabilities of Mathematica to represent abstract mathematics [closed]

I have an fundamental question about what Mathematica can and cannot do. I have a book which presents a certain physical theory in an axiomatic manner. The axioms make heavy use of mathematics. Some ...
• 333
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### Proving (or at least 'being told by Mathematica') that Sqrt[2] is irrational?

I realize that Mathematica is not specifically an automated theorem prover. However, this article: http://www.wolfram.com/products/mathematica/newin6/content/EquationalTheoremProving/ Suggests that ...
• 333
324 views

### Using Mathematica to confirm Bernoulli's inequality

I have several challenges that I want to confirm is true. I have chosen this one because it is rather simple (proof by induction). There are times when I do not want to spend ages trying find proofs. ...
• 1,209
140 views

### Performing an inductive proof

I have an expression which depends on a variable n and am certain it evaluates to 1 for all n with n=1/2,1,3/2,... I want to do an inductive proof that this is true but the actual formula is too ...
• 257
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### Can I define an axiomatic (Boolean algebra) system and prove theorems using Mathematica?

The general question is Can I define an axiomatic system and prove theorems using Mathematica? The more concrete one is about Boolean algebra. I consider this axiomatic Boolean algebra system (wiki)....
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### Proving the hairy ball theorem using xAct

I would like to formally prove the hairy ball theorem in Mathematica, initially just for $S^2$, and then see about generalizing. An approach I thought about to use the xAct package to define $S^2$ ...
• 1,065
1 vote
3k views

### Is this function convex?

How can I determine convexity of the function f = Log[ x, 1 + (x^a - 1) (x^b - 1)/(x - 1)] with the parameters $a,\,b$ belonging to the interval $(0,1)$ in ...
• 16.7k
411 views

### Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer.
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### Solving functional equations in Mathematica

I am trying to solve functional equations in Mathematica, but got nowhere. Just a simple example: $$f(x+y)^2=f(x)^2+f(y)^2$$ for all $x,y \in R$, assuming f is a real-valued function. How should I ...
337 views

### How can I prove an equation has a certain number of real roots?

I have this code ...
378 views

### Proving a recurrence in Mathematica

I have $$j_n=\int_0^1 x^{2n} \sin(\pi x)dx.$$ How do I show that $$j_{n+1}= \frac{1}{\pi^2}(\pi- (2n+1)(2n+2)j_n)\, ?$$ I keep getting a recurring integration by parts and I can't simplify it. ...
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### How to prove an expression by method of mathematical induction?

I want to prove the expression $$1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$$ by method of mathematical induction with Mathematica, but I do not know how to start. How do I tell ...
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### Proving (self) similarity with Mathematica - Reccurrence Plots, Similarity Plots etc

I posted this question in math.se but given the sheer tumultuous number of questions that keep appearing on math.se and also given that I am trying to accomplish this in Mathematica, I thought I'd ...
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### Using FindInstance to Prove No Solutions Exist

For a small amount of background, I am currently working on an undergraduate research project in Combinatorial Geometry and I'm working on a case analysis for embedding spherical simplicial 2-...
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