freckles one year ago $\text{ Let } m \text{ and } n \text{ be natural numbers such that } \\ A=\frac{(m+3)^n+1}{3m} \text{ is an integer } \\ \text{ Show } A \text{ is odd.}$

1. UsukiDoll

well definition of odd is 2k+1

2. freckles

Assume A is even then A=2i where i is an integer. $2i=\frac{(m+3)^n+1}{3m} \\ \text{ let's see if we can arrive at some contradiction } \\ 2i(3m)=(m+3)^n+1 \\ 2(3im)=(m+3)^n+1 \text{ so } (m+3)^n+1 \text{ is even } \\ 2(3im)-1 \text{ is odd } \text{ so } (m+3)^n \text{ is odd }$ thinking a bit more... (not sure if I'm going to get this to work)

3. freckles

having trouble with contradiction ...was thinking about writing it like this but I don't see how to use vieta's jumping of whatever it is called if I can even use that ) $A =\frac{(m+3)^n+1}{3m} \in \mathbb{Z} \\ A(3m)=(m+3)^n+1 \\ 0=(m+3)^n+1-A(3m) \\ (m+3)^n+1-3Am=0$

4. ganeshie8

The given problem is equivalent to : $\dfrac{m^n+3^n+1}{3m} =A \in \mathbb{Z} \implies \text{A is odd}$

5. freckles

you did mod 3

6. freckles

why does doing that given us an equivalent problem

7. ganeshie8

$(m+3)^n = m^n+3^n + 3m(\text{stuff})$

8. ganeshie8

Hmm yeah not exactly same as it disturbs the parity of $$A$$

9. freckles

actually found an answer but I don't like it because I don't understand it :p

10. freckles

do you want me to share it?

11. ganeshie8

yeah im almost stuck haha

12. freckles
13. freckles

click "This LTE thing is pretty cool"

14. freckles

where I have looked up LTE and I believe it just means lifting the exponent lemma/theorem thingy

15. freckles

if m is even why does that allow us to do mod 3?

16. freckles

By the way I got my problem from here if you wanted to know : https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w (by the way do you know if there is a way to hide how long this link actually is using some kind of LaTeX voodoo )

17. ganeshie8

click $$\href{https:///www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w}{this}$$

18. freckles

By the way I got my problem from here if you wanted to know : https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w (by the way do you know if there is a way to hide how long this link actually is using some kind of LaTeX voodoo )

19. UsukiDoll

20. freckles

omg too cute and it was A17 just in case you were scrolling for the probem

21. freckles

@UsukiDoll what is that "bit.ly sites"?

22. freckles

or how do you use it

23. ganeshie8

@freckles change http:// to http:/// 3 forward slashes are needed

24. UsukiDoll

bit.ly <--- online site used to shorten long url links

25. freckles

$\href{ http:///www.wolframalpha.com/input/?i=integrate%28kitty%2Ckitty%29 }{WOLFRAM!}$

26. ganeshie8

congrats!!

27. freckles

you can teach an old dog new tricks so congrats to you on doing just that :p

28. freckles

http://bit.ly/1SFjO6a this is the thing @UsukiDoll was talking about

29. freckles

That is kind of neat too

30. freckles

though I kind of like writing my links in cute words

31. UsukiDoll

bit.ly <--- online site used to shorten long url links

32. freckles

hmmm anyways a mod 3 =0,1, or 2 1 mod 3=1 2 mod 3=2 3 mod 3=0 4 mod 3=1 5 mod 3=2 6 mod 3=0 ... 1,4,...,1+3i mod 3 will give us 1 2,5,...,2+3i mod 3 will give us 2 3,6,...,3+3i mod 3 will give us 3 (which this makes sense) but any of 1+3i,2+3i,3+3i can be even or odd so I don't understand if we assume m is even why we can do mod 3 like what does it tell us doing that