# How is $(x+y)^p \equiv x^p + y^p$ mod $p$ for any prime number $p$?

I'm currently studying for an exam and on the practice test given to us (with solutions), there is a problem that states the following:

Notice that for all $$x,y \in \Bbb Z$$,

$$(x+y)^2 = x^2 + 2xy + y^2 \equiv x^2 + y^2$$ mod $$2$$

$$(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \equiv x^3 + y^3$$ mod $$3$$

$$(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 \equiv x^5+y^5$$ mod $$5$$

Prove that if $$p$$ is any prime number then for all $$x,y \in \Bbb Z$$ we have $$(x+y)^p \equiv x^p + y^p$$ mod $$p$$.

Here is the proof that was provided to us in the solutions:

Let $$p$$ be a prime number. If $$k$$ is an integer satisfying $$1 \leq k \leq p-1$$, then $$k! = 1\cdot2\cdot3\cdot\cdot\cdot k$$ is a product of positive integeres smaller than $$p$$, and therefore $$k!$$ is not divisible by $$p$$. For the same reason, if $$1 \leq k \leq p-1$$ then $$(p-k)! = 1\cdot2\cdot3\cdot\cdot\cdot (p-k)$$ is not divisible by $$p$$. Since $$p!$$ obviously is divisible by $$p$$, we infer that

$$\binom{p}{k} = \frac{p!}{k!(p-k)!} \equiv 0$$ mod $$p$$ whenever $$1 \leq k \leq p-1$$.

Therefore

$$(x + y)^p = x^p + y^p +\sum_{k=1}^{p-1} \binom{p}{k}x^k y^{p-k} \equiv x^p +y^p$$ mod $$p$$.

Q.E.D.

I'm having trouble understanding a few things about this theorem and proof.

First, I'm not quite seeing the congruence (if that makes any sense). Like, I'm not seeing how $$(x+y)^2 = x^2 + 2xy + y^2 \equiv x^2 + y^2$$ mod $$2$$, or rather just in general for any power $$p$$, not just $$2$$.

I recall that we say $$a \mid b$$ if there exists some $$c$$ such that $$ac = b$$, and that we say if $$a,b \in \Bbb Z$$ and $$n \in \Bbb N$$, $$a$$ is congruent to $$b$$ modulo $$n$$ if and only if $$n\mid (b-a)$$. This is written as $$a \equiv b$$ (mod $$n$$) where $$n$$ is called the modulus, and from this we get residue classes $$r$$ that are all the sets of numbers of the form $$a = bn + r$$ where $$r \in \{0,1,2,\cdot\cdot\cdot, n-1\}$$. I'm not quite sure how to phrase that but I can picture what it's supposed to be in my head.

I saw that in an answer listed here Proving $$(a+b)^{p} \equiv a^{p} + b^{p} \pmod{p}$$ for prime $$p$$ that someone says "because nothing in the denominator divides $$n$$. So all of the middle terms in the expansion of $$(a+b)^p$$ vanish modulo $$p$$, and you're left with the two ends: $$a^p+b^p$$." But it still isn't making sense to me. How is it that because the denominator doesn't divide the numerator, all the messy middle terms disappear?

The next question I have is why go about proving it this way? What is the motivation for the techniques used to prove this statement? When I first saw this problem, admittedly I didn't know where to start. Part of me thought about induction but that was quickly ruled out based on the fact we're only considering prime powers. I did however immediately recognize that we're using binomial expansion and pascal's triangle, but that's about it.

I apologize for the long winded post and lots of questions, but this problem currently isn't making a lot of sense to me, so I would be incredibly grateful if somebody could explain this proof and its ideas to me. Thank you!

• One preliminary observation is that $(a+b) \bmod{p} = ((a \bmod p) + (b \bmod p)) \bmod p$. This lets you remove the $2ab$ term from $a^2 + 2ab + b^2$. – Fabio Somenzi Sep 27 '18 at 2:58

## 3 Answers

The key here is to know that $(x+y)^n$ can be computed using the binomial expansion formula which explicit shows how the coefficients behave. Then it’s just a matter of seeing that each of those coefficients are divisible by $p$.

• Oh, okay, that's actually really helpful. Thanks! – Matthew Graham Sep 27 '18 at 3:07

By the binomial formula (and as the solution mentions), $$(x + y)^p=x^p +y^p + \sum_{k=1}^{p-1} \binom{p}{k}x^k y^{p-k}$$, and so $$x^p +y^p-(x + y)^p = -\sum_{k=1}^{p-1} \binom{p}{k}x^k y^{p-k}$$. Now we just need to show that there exists an integer $$c$$ such that $$pc = -\sum_{k=1}^{p-1} \binom{p}{k}x^k y^{p-k}$$. You should know that $$\binom{p}{k}=\frac{p!}{k!(p-k)!}$$ is an integer. Since $$p$$ is a prime, and as the solution mentions; if $$1 \leq k \leq p-1$$, then $$pd \neq k!$$ and $$pd \neq (p-k)!$$ for all integers $$d$$. So $$p$$ remains as a factor of $$\binom{p}{k}$$ (*informally, it cannot be "cancelled out" by $$k!(p-k)!$$). Hence $$c=-\sum_{k=1}^{p-1} \frac{1}{p}\binom{p}{k}x^k y^{p-k}$$ is an integer. Therefore, $$(x+y)^p$$ is congruent to $$x^p+y^p$$ modulo $$p$$.

*It would be helpful to investigate why the same argument does not apply in cases where $$p$$ is not a prime number, such as $$p=4$$ or $$p=6$$.

• In case something I said in my answer sounds dubious, I will note that $\neg \exists d \text{P}(d) \land \neg \exists d \text{Q}(d)$ is logically equivalent to $\forall d\left (\neg \text{P}(d) \land \neg \text{Q}(d)\right)$. – Matt A Pelto Sep 27 '18 at 4:41

If you are familiar with it, this problem follows immediately from Fermat Little Theorem. Indeed, by FLT you have $$(x+y)^p \equiv x+y \pmod{p} \\x^p \equiv x \pmod{p} \\y^p \equiv y \pmod{p}$$

therefore $$(x+y)^p \equiv x+y \equiv x^p+y^p \pmod{p}$$

• Oh that's nice! We haven't reached that theorem yet in my abstract algebra class, I'm sure we'll touch upon it soon, but I have seen that before in another class. Totally forgot though. Didn't think to use it. Thanks for the response! – Matthew Graham Sep 28 '18 at 22:32
• +1, but the general proof presented by the OP can be extended to computations in fields of nonzero characteristic, whereas this one cannot. – egreg Sep 28 '18 at 22:34
• @egreg Setting $z=xy^{-1}$ you need to show that $$(z+1)^p=z^p+1$$ this is a polynomial of degree at most $p-1$ which, by FLT, has all $p$ elements of the prime fields as roots. Therefore, it is the zero polynomial... How is that for an extension ;) – N. S. Sep 28 '18 at 22:44
• @N.S. :-) ............. – egreg Sep 28 '18 at 22:51
• @egreg Of course $(z+1)^p=z^p+1$ is equivalent to all middle coefficients being $0$, and this can also be used to prove the FLT.. So in some sense the Frobenius map being a morphism and FLT are equivalent :) – N. S. Sep 28 '18 at 22:54