If you already have a proof for some result but want to ask for a different proof (using different methods).
Questions tagged [alternative-proof]
3599 questions
20
votes
1 answer
Easy proof, that $\rm e\notin \mathbb Q$
$\def\e{{\rm e}}$
I recently had the task to explain the proof that $\e$ is irrational as a presentation to my classmates. To prepare this presentation, the teacher gave me a script with a proof that uses an estimation of the series $b_n =…
FUZxxl
- 9,307
16
votes
9 answers
Lesser known derivations of well-known formulas and theorems
What are some lesser known derivations of well-known formulas and theorems?
I ask because I recently found a new way to derive the quadratic formula which didn't involve completing the square as is commonly taught. Doing so I was wondering what…
Frank Vel
- 5,339
9
votes
7 answers
Elementary theorems with several proofs?
Every year my student's math club organizes a "proof marathon", where we present multiple proofs for a single theorem. For instance, last edition we did the AM-GM inequality with geometric, algebraic, analytic... proofs, and even one "proof" based…
Jens Bossaert
- 383
6
votes
3 answers
Cool property of the number $24$
Recently I've had my 24th birthday, and a friend commented that it was a very boring number, going from 23 which is prime, 25 which is the first number that can be written as the sum of 2 different pairs of squared integers $3^2+4^2 =0^2+5^2 =25$,…
Oria Gruber
- 12,739
5
votes
4 answers
How many triplet primes of the form $p, p+2, p+4$ are there? Prove your conjecture.
I am giving this problem to 8th grade students, and I am hoping that people can help me find elementary ways to prove this problem. I would love to find other arguments that are accessible to 8th graders, so that I can help them with their…
MathGuy
- 1,237
5
votes
3 answers
Proof of no primes such that $x^2 + y^2 = z^2$
I'm in a pretty simple "CS Math" course for year 1 Comp Sci, and I came across this:
Disprove, $x^2 + y^2 = z^2$, such that $x, y, z$ are primes
I thought of this as, if n is a prime, then prime factorization of n must be:
$n = z*1$
$n^2 =…
q.Then
- 3,080
4
votes
2 answers
Can you ever disprove a proof or prove something in more than one way
This seems to be a very simple question but surely because there is an infinite way of writing a=1 and b=2 in so many different forms, can you ever disprove a proof or prove something in more than one way. Thanks
jock214
- 71
3
votes
1 answer
Simple proof of some formula for n!
I have found an interesting identity for n! , but my proof is slightly complicated using Bernoulli numbers.
Can somebody find some simple proof of the following formula?
$$(-1)^n n!=\sum_{k=2}^{n+1}(-1)^{k+1}\binom{n+1}{k}\sum_{i=1}^{k-1}i^n,\quad…
Marek
- 695
- 3
- 14
3
votes
1 answer
Proving assertion with and without induction
I have successfully proven $ \displaystyle \sum_{k=1}^n k ·(k!) = (n+1)! -1 $ with mathematical induction for all $n \in \mathbb{N}$. Now, how would someone prove this assertion without induction?
steven2005
- 183
3
votes
1 answer
Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
My homework has this problem:
Prove that there exist irrational numbers $x$ and $y$ such that $x +
y$ is rational.
There is an easy solution that I found on mathbitsnotebook.com:
\begin{align} \left(2+6\sqrt{7}\right) + \left(-6\sqrt{7}\right) =…
Beginner
- 1,170
2
votes
4 answers
A more rigorous way to prove this?
I would like to prove the following statement
$$x^n-a^n=(x-a)\sum^{n-1}_{k=0}x^ka^{n-k-1},\qquad\forall n\in\Bbb N_0$$
I can easily prove it by induction using polynomial long division or series expansion however I am unsure whether or not these are…
Ali Caglayan
- 5,726
2
votes
4 answers
Better proof for $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
Prove this $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
My attempt
Proof - by using [axiomdistributive] and [axiommulcommutative]:
$$\begin{split}
&(x+y)(x^2 - xy + y^2)\\
&= (x+y)x^2 - (x+y)xy + (x+y)y^2\\
&= (x^3+x^2y) - (x^2y+xy^2) +…
caveman
- 393
- 1
- 13
1
vote
1 answer
Simplified Galois proof?
I have learned about Galois epoch-making proof that any polynomial of the fifth degree has no solution representable in terms of its coefficients.
Can his proof be simplified and clarified in modern language? If yes, please show that.
Yes
- 20,719
1
vote
2 answers
Consecutive positive integers proof problem
Consider any three consecutive positive integers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.
Work: I tried to prove via contradiction.
I made three integers, k, k+1, and k+2. Then set the equation$(k+2)^3 =…
mrQWERTY
- 595
1
vote
0 answers
Prove that in any set of ten different two-digit numbers one can select two disjoint subsets such that the sum of numbers in each subset is the same
Prove that in any set of ten different two-digit numbers one can select two disjoint subsets such that the sum of numbers in each of the subsets is the same.
My proof:
There are $2^{10} = 1024$ different subsets of the ten numbers.
Consider the…
