Oh my God!
What is that, Red Planet. What is it about, Science Fiction??
No way. That movie is way bad for me. I have never seen anything like that, which is called "SciFi".

I have seen bad SciFi movies, and the list is pretty long. I started with "Terminator 2" and felt pretty cool about it. I watched through Matrix and the like "alternate reality" movies, and found them great. However, we had been doing the "stereotyped Alien" for quite sometime now. We had been excusing the lack of imagination of the folks making it (alien is always almost like a human or an octopus or a lizard, nothing else). There is so much of variety of living beings on earth, don't these guys just change the "Alien" just once? And aliens are always pure evil, slimy things that kill for nothing.

And the stories never seem to end. Alien kills predator kills alien kills predator again, blah...blah...blah... In this movie, those people on the spaceship does not know that Mars has oxygen now, (just) enough so that, humans can breathe with no discomfort at all. And, the scientists who planted the algay and bugs in the first place on Mars, they did not know it. Surprise!!

For once, I got serious doubt on the ability of those Nasa scientists :-)

And, they went to Mars, just to find that their folks camp there is ransacked and everybody is killed. Another surprise!! Where are we now, don't we have phones and stuff to decide whether the base is in sane condition or not. Oh, for God's sakes, please find funds for Nasa, and it's Mars expedition. Too much over my head, and definitely no science in this, and not even good fiction.

And, thirdly, they have a bot, that can "search and destroy" aliens automatically. Since when is Nasa developing such a thing, really? I can imagine those scientists, who devoted their lives to find one living thing on Mars, and dreaming of capturing an alien and studying it to quench their thirsts of knowledge, but, I haven't heard of this breed of science folks, who had to make a bot, which automatically searches and destroys aliens. Is it that Nasa had come to this conclusion due to all Hollywood movies coming out currently.

I gave a moratorium on SciFi movies for myself, after watching it. My brain was so dead, that I could not watch any other SciFi (good or bad) for 2-3 months more.

Fermat's theorem about prime numbers

I could also prove the fermat's theorem about prime numbers,

(2^p - 1) == 1 (mod p) for any prime number. The proof uses binomial expansion.

sum of all nCr, r from 0 to n is 2^n. If we keep a prime number p for n,

pCr, r from 0 to p is 2^p.
In these co-efficients, except pC0, and pCp, everything else must have p as a factor, since, p is a prime.

So, finally, pC0 + pCp + pK = 2^p.
pK + 1 = 2^p - 1.

So, (2^p - 1) == 1 (mod p).

I was pleasantly surprised about this, although.
(This is called Fermat's little theorem)

The statement is more generic. If p is a prime, for any integer a, a^p == a (mod p).
This can be proved using induction, on a.

1^p == 1 (mod p) trivial, we have also seen 2^p == 2 (mod p) incidentally.
Also assume that n^p == n (mod p).
(n + 1)^p = Sum from 0 to p (pCr * n^r)

All terms except pC0 and pCp have p as the factor. So,

pC0 + pCp * n^p + pK = (n + 1)^p
n^p == n (mod p)
So,
pK + (n + 1) = (n + 1) ^p
So, (n + 1) ^p == (n + 1) mod p

Puzzle for guessing by not knowing

I came across this puzzle sometime back. There are two guys A and B. And there are two numbers x and y, which are between 2 and 9 (digits, 2 <= x <= y <= 9). A knows their sum, and B knows their product. And the dialogue between them goes as follows:

A: I don't know what the numbers are.
B: I don't know the numbers as well.
A: I still don't know what the numbers are.
B: Same, I don't know the numbers.
A: I still don't know the numbers.
B: I know the numbers now

What is the value of x + y?