help me people

My math teacher sent me some homework via EMAIL at YESTERDAY MIDNIGHT. And he wrote that me and another student has to solve all the tasks, or else we will be given enough grades to fail at the end of this term. ISN'T HE CRAZY???

So please, my fellows, help me shove the answers up his ass.
Here is the first i couldn't solve.

x^2 + 32x + 2264 = p^2; p is a prime of choice
find x, if x is an integer

tl;dr: please give tips how to solve the problem above

(x+16)^2 + 2008 = p^2
seems suspiciously like a competition question...

its okay, because we are being trained to solve competition questions...:D

>>1
Can't be done. >>2 is black.

but... after that..? i have to return integers

Actually 4 tran is a chink. from california.

>>1
I shall halp.

In either case, let y = x+16.  Then we can just solve for y and subtract 16 to get x.

y^2 + 2008 = p^2

(p-y)(p+y) = 2008

2008 = 8 * 251 (251 is prime)

p-y and p+y are both odd or both even.  Since you can't write 2008 as a product of odd numbers, they must both be even.  Therefore (p-y, p+y) must be either
(2, 4 * 251)
(4, 2 * 251)

If (a,b) is one of these pairs, then
p-y = a
p+y = b
and so p = (a+b)/2.

(2+4*251)/2 = 503 is prime, which gives the solution p=503, y=501 (and x=501-16=485).  However (4+2*251)/2 = 253 = 11*23 is not.  Therefore the only solution is

p = 503
x = 485

>>1
Post the rest here and I shall deliver the answers of correctitude.

>>7
Nice solution!  Why didn't I think of it?  Such phail.

In any case, you missed x = -517

well... these aren't putnam problems, cause i took the putnam two days ago, and that wasn't on there.

>>10
Not to mention that it's 5000% too easy for the putnam.

(If I hadn't blanked on A2 and B2 I would've gotten nine and coulda fukken won that cocksucker this year >:( .  I bet I'll be the only idiot with an honorable mention who DIDN'T get A2.  )

>>10
Even if they were putnam problems, he'd only have 3 hrs for us to poast an answer b4 failing.

>>11
Oh wow, 9 putnam problems.  I wish I were that smart.  The most I've managed to pull off was 4, and even then, I don't think I got anywhere near full credit.

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