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charybdis 2021-11-27 16:32

[QUOTE=R. Gerbicz;594026]...furthermore: k*binomial(p,k)=p*binomial(p-1,k-1) is also true...[/QUOTE]

And we can do this combinatorially too: both are the way of choosing k elements from a set of p elements, where we label one of those k elements as "special".
We get k*binomial(p,k) by choosing the set of k elements and then picking the special element, and p*binomial(p-1,k-1) by first picking the special element and then choosing the other k-1 elements from the remaining p-1 elements of our set.

More famously, binomial(p-1,k-1)+binomial(p-1,k)=binomial(p,k) is easiest to see combinatorially: we partition the binomial(p,k) choices of k elements from the numbers {1,...,p} into two, depending on whether they contain 1 or not. binomial(p-1,k-1) is the number of choices containing 1, and binomial(p-1,k) is the number of choices not containing 1.

Dr Sardonicus 2021-11-29 16:05

[QUOTE=R. Gerbicz;594026]Modified your idea, close to a pure combinatorial proof:

The k=0 case is trivial, so assume that 0<k<p,
it is known: binomial(p-1,k-1)+binomial(p-1,k)=binomial(p,k) [for here you don't need that p is prime]
<snip>[/QUOTE]Of course! And for 0 < k < p, [tex]\frac{p!}{k!(p-k)!}[/tex] clearly is divisible by p because the denominator is composed of factors less than p.

The result can be extended slightly. Since p divides binomial(p,k) for 0 < k < p when p is prime, we have the "freshman's dream" polynomial identity in [b]F[/b][sub]p[/sub][x,y]

[tex](x\;+\;y)^{p}\;=\;x^{p}\;+\;y^{p}[/tex]

Repeatedly raising to the p[sup]th[/sup] power, we see that in [b]F[/b][sub]p[/sub][x,y] for any positive integer n,

[tex](x\;+\;y)^{p^{n}}\;=\;x^{p^{n}}\;+\;y^{p^{n}}[/tex]

which shows that binomial(p[sup]n[/sup],k) is divisible by p for 0 < k < p[sup]n[/sup].

Then the above argument shows that for p prime and any positive integer n,

[TEX]{p^{n}-1\choose k}\equiv -1^k \;\pmod p[/TEX]

MattcAnderson 2021-12-03 13:29

more of this
 
I am excited about this.
*smile*

bur 2021-12-10 17:33

Thanks a lot for all the replies, it will take me while to get through all that.


By "reduced" I meant, there would be no smaller integer congruent to them mod p. I forgot about that smaller could also mean "negative".

bur 2022-01-31 18:35

And another one:


An exercise deals with an integer a having order 3 mod p. And it is to be shown that a+1 has order 6 mod p.

As a hint they give: [TEX]a^2 + a + 1 \equiv 0 \pmod {p}[/TEX]. If I accept that, I can prove that a+1 will have order 6. But how do I get from [TEX]a^3 \equiv 1[/TEX] to [TEX]a^2 + a + 1 \equiv 0[/TEX]?

My problem is that I don't know what I can transform the a^3 into. [TEX]a^4 \equiv a[/TEX], but does that help? I also tried writing it as [TEX]a^3-1 = r*p[/TEX] which also didn't get me anywhere.

A small nudge in the right direction would be appreciated.

charybdis 2022-01-31 19:35

Hint: how does a^3-1 factorize?

R. Gerbicz 2022-02-01 11:41

[QUOTE=bur;599143]And another one:

An exercise deals with an integer a having order 3 mod p. And it is to be shown that a+1 has order 6 mod p.

As a hint they give: [TEX]a^2 + a + 1 \equiv 0 \pmod {p}[/TEX]. If I accept that, I can prove that a+1 will have order 6. But how do I get from [TEX]a^3 \equiv 1[/TEX] to [TEX]a^2 + a + 1 \equiv 0[/TEX]?[/QUOTE]

That is a wrong way and it is not even true: let a=1 (and p>3) then a^3=1 mod p but a^2+a+1 != 0 mod p.

The idea is only that, if you need just that order of (a+1) divides 6, then:

(a+1)^6==1 mod p, but that is true, because
a^2+a+1 | (a+1)^6-1 for every a integer, here a^2+a+1 is divisible by p, so (a+1)^6-1 is also divisible by p, hence (a+1)^6==1 mod p.

You need more, the order is 6, so it can't be 1,2 or 3 [we know that every "good exponent" is divisible by the order, so the order should divide 6]. If the order is odd, then:
(a+1)^3==1 mod p
in this case (a+1)^3-1-(a^2+a+1)*(a+2)=-2 but the left side is divisible by p, because (a+1)^3-1 and (a^2+a+1) is divisible by p.
This would mean that -2 is divisible by p, but p=2 can't be.

Now you need to handle only the order=2 case, so you can't have (a+1)^2==1 mod p. I left it for you.

Dr Sardonicus 2022-02-01 12:24

[QUOTE=bur;599143]And another one:

An exercise deals with an integer a having order 3 mod p.
<snip>[/QUOTE]This means that a^3 == 1 (mod p) but a^1 =/= 1 (mod p); that is, a^3 - 1 == 0 (mod p) but a - 1 =/= 0 (mod p).

Supplementary exercise: If p is prime, n divides p-1, and a^n == 1 (mod p), then the order of a (mod p) is exactly n

if and only if a^d =/= 1 for any divisor d of n with n/d > 1.

Refinement: If and only if a^(n/q) =/= 1 (mod p) for every prime factor q of n


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