5^x ≅ 1 (mod 4) My mind was blown when I produced this for

more generally you have a^x = b^x mod a-b

>>8850494

yes, you create the modulus by subtracting residues

I came about it form this problem:
prove 4 evenly divides 5^n - 1
>prove 4 | 5^n -1

I love modular arithmatic. I proved a general equation that determines if a 'evenly divisible' statement is true or not. it must be in this form to be true:
>M | (m+r)^n - (m-(m-r))^n

this must be true for all integers m,n and for all natural numbers n.

4 | 5^n -1 ∵ m=4,r=1
>4 | (4+1)^n - (4-(4-1))^n
>4 | (5)^n - (1)^n
>4 | 5^n - 1
∴ true

>>8850494
>a^x = b^x mod a-b
I like that

[math] n^{x+1} = n \cdot n^{x} [/math]
[math] = n^{x} + n^{x} +\ ...\ +n^{x} [/math]

By the linearity of mod,
if
[math] n^{x} = 1 [/math] mod [math]m[/math],
then
[math] n^{x+1} = n [/math] mod [math]m[/math].

Of course, if [math]m[/math] is one smaller than [math] n [/math], then
[math] n[/math] mod [math]m[/math]
is [math]1[/math] again.
And by induction it holds for any [math]x[/math].

[eqn]5^x = 1 + 4 \sum_{k=1}^x {x \choose k} 4^{k-1} [/eqn]

File: snipple.png (22KB, 764x247px) Image search: [Google]

22KB, 764x247px

>>8850491
read a book about number theory you putrid retarded piggot (pig + faggot)

>>8850529
modulus wasnt prime tho, and this needed to work for every p, not just p's that are not multiples of k

>>8850529
but that theorem has nothing to do with what he posted

>>8850491
>5^x ≅ 1 (mod 4)
my math teacher doubts this. she doesnt think its true.
she actually wrote that my proof was wrong because she didnt think that was correct.

I have one stupid math teacher

>>8850520
That's also a very nice way to approach it.

We may point out

[math] a^{n+1}-b^{n+1} = (a-b) \sum_{k=0}^n a^k\,b^{n-k} [/math]

resp.

[math] a^{n+1}-b^{n+1} = (a-b)\prod_{k=1}^n (a - b \cdot e^{ 2\pi i \ \frac{k} {n+1} } ) [/math]

(the latter being "evident" if you can use the fundamental theorem of algebra)

and set a=5, b=1

>>8850520
I like this method

>>8850529
Did the rich kids bully you again today, anon?There, there, you go and put that dumb piggot in his place.

>>8850538
>she didn't think it was correct
how the fuck does someone that retarded land a position in education?

File: piggot.png (114KB, 775x678px) Image search: [Google]

114KB, 775x678px

>>8850534
>>8850533
>>8850561
>fermat's little theorem is unrelated to blah blah blah
brainlet piggots (pig + faggot) detected

pic related is a generalization of the previous pic, piggots

>>8850575
California university, my friend.
I actually am planning on going by her office with a proof of just that one line

>>8850581
you've now posted two theorems that have nothing to do with what OP posted

are you just posting random things from babby's first elementary number theory text? do you even know what the theorems say?

>>8850529
>tfw I am taking a number theory class with a professor who always says he will do the proof from an algebraic perspective instead of from a number theory perspective

God fucking damn now I understand him. Look at that disgusting computation. DOT DOT DOT AM I RITE GUISE XD.

God damn, number theory was a mistake. I am so happy I was born in a time when algebra already saved the day and I can prove all theorems with group theory and I can just use pure number theoretical arguments when I need to solve problems.

Like seriously, the algebraic proof for this is so beautiful and straight forward.

>>8850586
I think you don't. Piggot detected.

What's worse is that you humiliated yourself by saying that the theorem you don't understand is elementary. You're a brainlet and I proved this by letting you prove it.

>>8850590
>What's worse is that you humiliated yourself by saying that the theorem you don't understand is elementary.
you just exposed that the only number theory you know is elementary number theory

nice work brainlet.

>>8850587
Man why are you writing in all caps no need to yell. Rewrite your reply and i will read it then.

>>8850603
Not true, brainlet. Me mentioning a basic theorem in number theory does not mean that this is all I know. How did your brainlet mind come to that conclusion?

Is all my vocabulary represented only by the words that I have written here in this post? That's absurd, you brainlet.

>>8850701
i called the theorem a theorem of elementary number theory and the fact you didn't recognize this as such makes it clear you're not aware of anything past the most basic of number theory

try again brainlet, perhaps when you've opened a book or two on analytic number theory, algebraic number theory, arithmetic geometry and modular forms.

i'll be waiting.

>>8850587
hey i can use caps too

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>>8850714
OP started a thread about a very basic concept in number theory concerning the prime number 5 and the composite 4

what did you expect to find in this thread, honestly?

you brainlets are amazing

>>8850734
>what did you expect to find in this thread, honestly?
not brainlets spouting off unrelated theorems like fermat's theorem or euler's theorem

>>8850743
I didn't expect to find in this thread brainlets who did not understand basic number theory but I did. What can I do? Such is life. Brainlets are everywhere, even on /sci/.

Perhaps first try to understand basic number theory before coming and enumerating random fields that employ number theory like a defective printing machine.

>>8850752
who doesn't understand basic number theory here other than you? everyone seems to understand OPs result except you posting unrelated results

>>8850755
Wow. No sense in arguing with you anymore. Now you're starting to directly use ad populum arguments to insult me.

The point is that you don't understand basic number theory and no amounts of insults can change that, brainlet. You don't even understand what a fucking prime number is and what is the relationship between composites and primes and why 5 behaves the way it does in OP's example.

You just said "oh it's true cause of the rules". You don't analyze the rules you just eat it up like a stupid piggot and assume it's true instead of trying to understand it. You don't understand shit. End of discussion.

>>8850764
>. You don't even understand what a fucking prime number is and what is the relationship between composites and primes and why 5 behaves the way it does in OP's example.
OPs example requires no use of primality. (note that none of the first three valid explanations/variants before you posted the irrelevant fermat theorem mentioned primes)

try again brainlet.

>>8850764
looks like that b8 worked well m8
another Sunday well spent
enjoy your >>>/h/
you've earned it

>>8850786
enjoy your /sci/ it;s the same thing, piggot

>>8851110
Not quite. /h/'s community is slightly more intelligent

>>8851199
digits confirm