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The Proof is Trivial! Watch

1. (Original post by j.alexanderh)
Solution 7:

Spoiler:
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since is prime.

Now

If we have the solution

It is easily verified that there is no solution if

If ,

which clearly has no solutions.

Hence the only solution is
Haven't seen you around in a while! How have you been?
2. Problem 9**

Show that there is no function satisfying
3. (Original post by und)
Haven't seen you around in a while! How have you been?
I'm pretty good, thank you. Et toi?

Problem 10*:

Evaluate
4. Solution 10

Hence
5. (Original post by Lord of the Flies)
Solution 10

Hence
Dat substitution
6. Solution 8

Let run from to and is the prime number .

Consider where is the highest power occuring in the prime factorisations of all possible values of

When is not a prime power has a factor of the form where are the primes in the prime fact. of When is a prime power we need to check that is not a power of . This is clearly true since is a multiple of and thus is not
7. (Original post by Lord of the Flies)
Problem 9**

Show that there is no function satisfying
Solution 9

Looking at the sets and clearly . If such an exists, it is injective and is also bijective between the sets and so contains an even number of elements, which is contradicted by the fact the cardinality of is
Let me try again.

Solution 6

Secondly, by induction, we obtain . Consequently, is a directed complete partial order, and its supremum is obviously .
Excellent. However the steps after this point can also be completed (more cleanly I think) without the use of contradiction.

Consider any satisfying . Since we have . Hence by induction for any natural and therefore any satisfying is an upper bound for the CPO . However out of all of them is the supremum of the CPO, so by definition the smallest.

The point of the question was to get you to prove an instance of Kleene's fixed-point theorem.
9. (Original post by Lord of the Flies)
Solution 10

Hence
Concise, as usual.
10. (Original post by Noble.)
...
11. Am I allowed to bombard this thread with problems?
12. Problem 11* (if you have never seen substitution before, **)

Find the general solution, by a suitable substitution or otherwise of this differential equation.

13. (Original post by shamika)
Am I allowed to bombard this thread with problems?

(Original post by j.alexanderh)
I'm pretty good, thank you. Et toi?
I'm good. Applying for maths at university I take it?
14. Problem 12*

I have a cube and I want to paint each side a different colour. How many different ways can I paint the cube with colours, ?
15. (Original post by Star-girl)
Problem 12*

I have a cube and I want to paint each side a different colour. How many different ways can I paint the cube with colours?
Solution 12

First we choose the six colours, so there are n choose 6 possibilities . Without considering identical cases we have possibilities for colouring the cube, but we divide by where 6 is the number of possible anchor faces and 4 is due to rotation about the anchor face, so we get
16. (Original post by Star-girl)
Problem 12*

I have a cube and I want to paint each side a different colour. How many different ways can I paint the cube with colours?
A cube has 6 sides and there are 24 ways to orientate a cube (each one of six faces can be placed and each placement can be rotated about 0, 90, 180 or 270 degrees). So the number of distinct cubes that are possible if we consider each side to be different is 6!/24 = 30. Given n colours, we must choose 6 (one for each face). So overall we have 30*nC6

(I assumed that n>6).

Problem 13*

Find the exact value of:

As an extention, try the above with 6 instead of 42. Also try 2 and then try 8. Can you spot a pattern and/or find a way to spot more numbers that "work"?
17. Solution 13

which has solutions for all positive n (pick the positive one)

The required solution is an integer when (the solution is then k).
18. Problem 14**

Prove that is divisible by 61.
19. (Original post by Star-girl)
Problem 14**

Prove that is divisible by 61.
Solution 14

As 61 is prime and 2 is not divisible by 61

by Fermat's Little Theorem. Hence:

20. Problem 15*/**

Evaluate

What if the upper limit in the integral were fixed, say

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