Goldbach conjecture, twin prime conjecure

Goldbach conjecture is one of the oldest unsolved problems in number theory.It states:
Every integer greater than 2 can be written as sum of 2 primes.
Here are some examples:6=3+3;12=7+5;32=29+3...
Twin prime conjecture states that there are infinitely many twin primes.
Examples:3,5;5,7;11,13...
I`ve been trying to solve these two, but you can`t solve them with primary school education.Although they seem simple, these are one of the hardest problems I tried to solve.

Fermat last theorem

Prove Fermat's Last theorem for n=3 : X^3 + Y^3 = Z^3 where X, Y, Z are rational integers, then X, Y, or Z is 0.
This is interesting one.Here`s something about Fermat from wiki:
is the name of the statement in number theory that:

It is impossible to separate any power higher than the second into two like powers,

or, more precisely:

If an integer n is greater than 2, then the equation aⁿ + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c.

In 1637 Pierre de Fermat wrote, in his copy of Claude-Gaspar Bachet`s translation of the famous Arithmetica of Diophantus, "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.")

Fermat's Last Theorem is strikingly different and much more difficult to prove than the analogous problem for n = 2, for which there are infinitely many integer solutions called Pythagorean triples (and the closely related Pythagorean theorem has many elementary proofs). The fact that the problem's statement is understandable by schoolchildren makes it all the more frustrating, and it has probably generated more incorrect proofs than any other problem in the history of mathematics. No correct proof was found for 357 years, when a proof was finally published by Andrew Wiles in 1995. The term "last theorem" resulted because all the other theorems proposed by Fermat were eventually proved or disproved, either by his own proofs or by other mathematicians, in the two centuries following their proposition. Although a theorem now that it has been proved, the status of Fermat's Last Theorem before then, in spite of the name, was that of a conjecture, a mathematical statement whose status (true or false) has not been conclusively settled.

Fermat's Last Theorem is the most famous solved problem in the history of mathematics , familiar to all mathematicians, and had achieved a recognizable status in popular culture prior to its proof.

USA math olympiad problem

Show that for any fixed integer n >=1, the sequence: 2, 2^2, 2^(2^2), 2^(2^(2^2))... mod n $$
is eventually constant. [That is, the sequence is defined: a_1=2, ai+1 = 2a_i. and you need to show that, for any positive integer n, the sequence a_1 mod n, a_2 mod n, .... is eventually constant.]

More number theory

Number theory is my favorite, and most of these problems are about numbers.Here`s another one:
if 11 divides (a+13*b) and 13 divides (a+11*b)
what is the least value of (a+b)?
This one`s easy, so here`s solution:
a+13b=11r =>a+2b=11p, where p=r-b
a+11b=13s =>a-2b=13q, where q=s-b
2a=11p+13q,
4b=11p-13q, thus p>q.
The least solution is: p=3 and q=1, a=23 and b=5, a+b=28,

Another math problem

If k*(a*b+1)=a^2+b^2, prove that k=n^2.
This one is very hard, so sharpen your pencils.

Learning math

Best and only way to learn math is to practice, practice and practice.Try solving some math problems here and you`ll see the difference.
A few simple math problems:
2 cats eat 2 mice in 2 days.How many cats will eat 100 mice in 100 days?

How much is 2+2/2-2*2/2-2?

How much is 2^2007 / 2^2008?(note: 2^3=2*2*2)

Is 2^2007+1 prime?

Number theory

I have been practiced number theory, so expect some number theory problems.

Power of 2 , another math problem

If 2^23 + 2^22 + ... + 2^14 + 2^13 - (x/5+6) is divisible by 1755, find x.
This one is mine too.
It`s similar to my first math problem, so if you can solve that, you can solve this.

Another math problem

Create two six-sided dice, such that the probability of each sum from 2 to 12 is the same as two standard dice. Each side must have at least one dot. Negative numbers are not allowed. There is another answer besides two standard {1,2,3,4,5,6} dice.

Try solving this!

Math Jokes

Here are some jokes I`ve stumbled upon recently:

Teacher: What is 2k + k?
Student: 3000!

Q: How does one insult a mathematician?
A: You say: "Your brain is smaller than any >0!"

A woman in a bar tries to pick up a mathematician.
"How old, do you think, am I?" she asks coyly.
"Well - 18 by that fire in your eyes, 19 by that glow on your cheeks, 20 by that radiance of your face, and adding that up is something you can probably do for yourself..."

Life is complex: it has both real and imaginary components.

Q: How does a mathematician induce good behavior in her children?
A: `I've told you n times, I've told you n+1 times...'

An investment firm is hiring mathematicians. After the first round of interviews, three hopeful recent graduates - a pure mathematician, an applied mathematician, and a graduate in mathematical finance - are asked what starting salary they are expecting.
The pure mathematician: "Would $30,000 be too much?"
The applied mathematician: "I think $60,000 would be OK."
The math finance person: "What about $300,000?"
The personnel officer is flabberghasted: "Do you know that we have a graduate in pure mathematics who is willing to do the same work for a tenth of what you are demanding!?"
"Well, I thought of $135,000 for me, $135,000 for you - and $30,000 for the pure mathematician who will do the work."

Many more at math.ualberta.ca/~runde/jokes.html.

The Riemann hypothesis

The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics.It has been an open question for almost 150 years, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs.

The Riemann hypothesis (RH) is a conjecture about the distribution of the zeros of the Rieman zeta function ζ(s). The Riemann zeta-function is defined for all s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, s = −4, s = −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:

The real part of any non-trivial zero of the Riemann zeta function is ½.

Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function whose real zeros correspond to the zeros of the zeta-function on the critical line.

The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true. A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.Wow.

My first mathematics problem

This one is mine:
Prove that (2^2008 + 2042)^3 is divisible by 511.
2^3=2*2*2.
I`ll post solution later.

A few interesting math problems - JBMO 2007

Here are problems from Junior Balkan MO 2007:

1	Let be positive real number such that . Prove that the equation has no real solution.	S
2	Let be a convex quadrilateral with , and . The diagonals and intersect at point . Determine the measure of .	S
3	Given are points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least scalene triangles with vertices of that color.	S
4	Prove that if is a prime number, then is not a perfect square.

These are not very hard, but it took me 2 hours to solve 3. one, cuz I`m not very good at analysis.
Here are solutions to all of them(click on here :)).

Hello

I`ve started this blog, and I`ll post here some interesting math problems and interesting facts about mathematics.