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amazing facts

amazing facts

1. There are 62,000 miles of blood vessels in the human body – laid end to end they would circle the earth 2.5 times

2. At over 2000 kilometers long, The Great Barrier Reef is the largest living structure on Earth

3. The risk of being struck by a falling meteorite for a human is one occurrence every 9,300 years

4. A thimbleful of a neutron star would weigh over 100 million tons

5. A typical hurricane produces the energy equivalent of 8,000 one megaton bombs

6. Blood sucking hookworms inhabit 700 million people worldwide

7. The highest speed ever achieved on a bicycle is 166.94 mph, by Fred Rompelberg

8. We can produce laser light a million times brighter than sunshine

9. 65% of those with autism are left handed

10. The combined length of the roots of a Finnish pine tree is over 30 miles

11. The oceans contain enough salt to cover all the continents to a depth of nearly 500 feet

12. The interstellar gas cloud Sagittarius B contains a billion, billion, billion liters of alcohol [JFrater is planning to move there in the near future]

13. Polar Bears can run at 25 miles an hour and jump over 6 feet in the air

14. 60-65 million years ago dolphins and humans shared a common ancestor

15. Polar Bears are nearly undetectable by infrared cameras, due to their transparent fur

16. The average person accidentally eats 430 bugs each year of their life

17. A single rye plant can spread up to 400 miles of roots underground

18. The temperature on the surface of Mercury exceeds 430 degrees C during the day, and, at night, plummets to minus 180 degrees centigrade

19. The evaporation from a large oak or beech tree is from ten to twenty-five gallons in twenty-four hours

20. Butterflies taste with their hind feet, and their taste sensation works on touch – this allows them to determine whether a leaf is edible

The Difference Between Insects And Spiders ?

The Difference Between Insects And Spiders ?
Despite the visual similarities between the two both are actually members of distinct families. Spiders are members of the Arachnid family whilst insects belong to the Insect family. So what makes the two families different ?
Well for starters spiders have 8 legs whilst insects have just 6. Insects have 3 distinct body parts, a head, thorax and abdomen whilst spiders have just 2, a combined head and thorax known as the cephalothorax and an abdomen.
As well as the above, other interesting differences include the eyes. Spiders have simple eyes whilst insects have compound eyes. Spiders have piercing jaws whilst insects have jaws which are more suitable for chewing. All spiders can make silk thread (although not all spiders spin webs) whilst most insects cannot. Spiders can’t fly where as many insects have wings allowing them to do so.
So yes to the untrained eye there is indeed very little visual difference but if you look really hard then they’re actually worlds apart.

kepler's 1st law proff

Kepler's first law
Now that we have polar basis vectors under our wing (and the polar representations of velocity and angular momentum), we are ready to proceed with the proof of Kepler's first law -- that the orbits of planets are ellipses with the Sun at one focus.To begin with, we will start off by applying Newton's law of motion and Newton's law of universal gravitation together to find that
m a = (-G m M/r2) r	equation 1
and, dividing both sides of the equation by m,
a = (-G M/r2) r.	equation 2
Recalling our work with polar basis vectors, we know dtheta/dtheta = -r. Solving for r and applying the chain rule, we find that
r = -dt/dtheta dtheta/dt	equation 3
r = -1/omega dtheta/dt.	equation 4
Substituting this into our equation for a we find
a = (-G M/r2) (-1/omega) dtheta/dt	equation 5
a = (G M)/(r2 omega) dtheta/dt.	equation 6
If we multiply the right side of the equation by m/m (which is unity), we obtain
a = (G m M)/(m r2 omega) dtheta/dt.	equation 7
But l = m r2 omega (I told you this would come in handy as well), so we can rewrite this as
a = (G m M/l) dtheta/dt.	equation 8
We can multiply both sides by l/(G m M) and find
l/(G m M) a = dtheta/dt.	equation 9
But we know that a = dv/dt, and can substitute accordingly:
l/(G m M) dv/dt = dtheta/dt.	equation 10
This is a differential equation that we can now solve. Upon solving it, we find that
l/(G m M) v = theta + C	equation 11
where C is some constant vector. We'll solve this for v and find that
v = (G m M/l) (theta + C).	equation 12
This is a general solution to the differential equation.But we're not finished. This doesn't tell us much about the shape of a planet's orbit, although all the pieces are there. This is the general solution, and it could be an orbit of any of the possible shapes (though we can't be sure what they are yet) or any of the possible orientations. We're interested in knowing the shape, of course, so we want to restrict the possible orientations. To do that, we'll take a special case. It makes sense to have perihelion -- that is, closest approach to the Sun -- at time t = 0. We'll restrict the orientation so that, when perihelion occurs, the planet lies along the zero radian line from the Sun (or, in Cartesian terminology, along the positive x-axis) -- that is, theta = 0. At this point, r, the position vector of the planet, will have only a component in the positive x-axis. We'll also assume that the planet orbits the Sun counterclockwise, through increasing measures of angles. If this is the case, then the velocity v at the instant of perihelion should be orthogonal to the position vector r, and it should have only a component in the positive y-axis. According to our expression for v, we have a scalar times the vector quantity theta + C.
theta|(theta = 0) = j;	equation 13
that is, the unit transverse vector points "up" when the unit radial vector points "right." Since, at t = 0, theta points entirely in the y-direction, then our constant vector C must only have a component in the y-axis -- this is the only way to get a resultant vector (v) that points entirely in the y-direction. So, we can rewrite C as a scalar times the unit basis vector in the y-direction:
C = e j	equation 14
where e is some scalar constant. (You will find that we have a very good reason for choosing the letter e in this case.) Substituting this into our equation for v, we get
v = (G m M/l) (theta + e j).	equation 15
This is the specific case when we want the orbit oriented so that perihelion occurs at t = theta = 0.Now we are ready to finish up the problem. We can dot both sides of the equation with theta and get:
v dot theta = (G m M/l) (theta + e j) dot theta	equation 16
v dot theta = (G m M/l) [theta dot theta + e (j dot theta)].	equation 17
A vector dotted with itself yields the square of that vector's magnitude, so theta dot theta = 1. Simplifying v dot theta, we find
v dot theta = (dr/dt r + r omega theta) dot theta	equation 18
v dot theta = dr/dt (r dot theta) + r omega (theta dot theta).	equation 19
But the dot product of two orthogonal vectors is zero, so r dot theta = 0. We also already know that theta dot theta = 1. Therefore
v dot theta = r omega.	equation 20
The last part of our problem is finding an expression for j dot theta. We know that theta = -sin theta i + cos theta j (by definition), so
j dot theta = j dot (-sin theta i + cos theta j)	equation 21
j dot theta = cos theta.	equation 22
We have the three pieces of the puzzle, so we'll put them together to find that
r omega = (G m M/l) (1 + e cos theta).	equation 23
We see r omega on the left side of this equation. We know that l = m r2 omega, and it would be nice to get rid of the mess that way. So we'll multiply both sides of the equation by m r to get
m r2 omega = (G m2 M/l) [r (1 + e cos theta)].	equation 24
Replacing the left side of the equation by l and moving the constants to the left side of our equation, we find that
l = (G m2 M/l) [r (1 + e cos theta)].	equation 25
We're almost finished now. Here it is clear that we have an explicit function in terms of r and theta -- in other words, this should be the polar equation for our planet's orbit! All we need do now is solve for r:
r = [l2/(G m2 M)]/(1 + e cos theta).	equation 26
The equation of a conic section with focus-directrix distance p and eccentricity e is represented by the polar equationa
r = e p/(1 + e cos theta).	equation 27
But this is exactly what we have, given that
e p = l2/(G m2 M).	equation 28
The focus-directrix distance should be a constant, which p is: l, G, m, and M are all individually constant; therefore the expression l2/(G m2 M) must also be constant. Therefore, Newton's laws of motion and universal gravitation dictate that the orbits of planets follow conic sections. This is Kepler's first law.... Well, almost. Kepler's first law actually states that planets follow the paths of ellipses. An ellipse is only one type of conic section. So why is an ellipse allowed while the others are not? Because the others are allowed -- we just don't call them planets. When Kepler said planet, he meant a body which returns to our skies over and over again. This means that the curve representing the orbit must be closed -- that is, it must continue to retrace itself over and over. The only two conic sections which are closed are the circle and the ellipse (the circle is just a special case of the ellipse anyway). Thus, any body in the skies we see over and over must be orbiting the Sun in an ellipse. The other two conic sections -- the parabola and hyperbola -- are open curves and correspond to a position where the body has sufficient velocity to escape from the Sun's gravity well. The body would approach the Sun from an infinite distance, round the Sun rapidly, and then recede away into the infinite abyss, never to be seen again. So we have proved an extension of Kepler's first law: A body influenced by the Sun's gravity follows a path defined by a conic section with the Sun at one focus.

boolos quesion

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

^{it is also provided}

• It could be that some god gets asked more than one question (and hence that some god is not asked any question at all).
• What the second question is, and to which god it is put, may depend on the answer to the first question. (And of course similarly for the third question.)
• Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely. 

• Random will answer da or ja when asked any yes-no question.

Boolos provided his solution in the same article in which he introduced the puzzle. Boolos states that the "first move is to find a god that you can be certain is not Random, and hence is either True or False". There are many different questions that will achieve this result. One strategy is to use complicated logical connectives in your questions (either biconditionals or some equivalent construction).

Boolos' question was to ask A:

Does da mean yes iff you are True if B is Random?

Equivalently:

Are an odd number of the following statements true: you are False, da means yes, B is Random?

It was observed by Roberts (2001) -- and independently by Rabern and Rabern (2008) -- that the puzzle's solution can be simplified by using certain counterfactuals. The key to this solution is that, for any yes/no question Q, asking either True or False the question

If I asked you Q, would you say ja?

results in the answer ja if the truthful answer to Q is yes, and the answer da if the truthful answer to Q is no (Rabern and Rabern (2008) call this result the embedded question lemma). The reason it works can be seen by looking at the eight possible cases.

• Assume that ja means yes and da means no.

1. True is asked and responds with ja. Since he is telling the truth, the truthful answer to Q is ja, which means yes.
2. True is asked and responds with da. Since he is telling the truth, the truthful answer to Q is da, which means no.
3. False is asked and responds with ja. Since he is lying it follows that if you asked him Q he would instead answer da. He would be lying, so the truthful answer to Q is ja, which means yes.
4. False is asked and responds with da. Since he is lying it follows that if you asked him Q he would in fact answer ja. He would be lying, so the truthful answer to Q is da, which means no.

• Assume ja means no and da means yes.

1. True is asked and responds with ja. Since he is telling the truth, the truthful answer to Q is da, which means yes.
2. True is asked and responds with da. Since he is telling the truth, the truthful answer to Q is ja, which means no.
3. False is asked and responds with ja. Since he is lying it follows that if you asked him Q he would in fact answer ja. He would be lying, so the truthful answer to Q is da, which means yes.
4. False is asked and responds with da. Since he is lying it follows that if you asked him Q he would instead answer da. He would be lying, so the truthful answer to Q is ja, which means no.

The solution below constructs its three questions using the lemma described above.

Q1: Ask god B, "If I asked you 'Is A Random?', would you say ja?". If B answers ja, then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is indeed Random. Either way, C is not Random. If B answers da, then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is not Random. Either way, A is not Random.

Q2: Go to the god who was identified as not being Random by the previous question (either A or C), and ask him: "If I asked you 'Are you True?', would you say ja?". Since he is not Random, an answer of ja indicates that he is True and an answer of da indicates that he is False.

Q3: Ask the same god the question: "If I asked you 'Is B Random?', would you say ja?". If the answer is ja then B is Random; if the answer is da then the god you have not yet spoken to is Random. The remaining god can be identified by elimination.

Diophantine equations

Diophantine equations 
There are two main topics that everyone who has any knowledge of number theory is aware of: prime numbers and Diophantine equations. Prime numbers are learned about early in the study of elementary arithmetic, but there are deep and fascinating questions connected with their distribution, and an elegant existing theory dealing with such questions, so let's defer that. Diophantine equations, on the other hand, provoke a multitude of special questions, but relatively little organized theory has developed to deal with them.
Diophantine equations get their name from Diophantos of Alexandria. Diophantos flourished in the 3rd century CE and wrote a highly regarded treatise he called the Arithmetica. Much of this dealt with solving algebraic equations in several unknowns, with the added restriction that solutions had to be rational or integral. This was especially important to the Greeks because of the discovery by Pythagoreans centuries earlier that many solutions of algebraic equations, even something as simple as the square root of 2, were not rational numbers. Then, as now, the term "irrational" had very negative connotations (though no longer in modern times as regards numbers).
In any case, early mathematicians were very interested in finding solutions of simple algebraic equations that are rational or integral numbers. This is often a very natural condition to impose on a particular problem. The equation, for example, may apply to things which ordinarily occur in whole number amounts, such as living animals.
A Diophantine equation, then, is an algebraic equation for which rational or integral solutions are sought. (Systems of such equations are also considered.) An algebraic equation is one that involves only polynomial expressions in one or more variables. What makes the equation "Diophantine" is that the coefficients of the polynomials should be rational numbers (or often, integers), and further that only rational (or integer) solutions are sought. Since these are the only conditions, not much in the way of a general theory has developed, though a lot is known about many more specialized cases.
The closest thing to a general theory is to be found in algebraic geometry, which deals with the geometric properties of solution sets of algebraic equations, usually over an "algebraically closed" field of numbers, such as the complex numbers. Algebraic geometry, indeed, turns out to be very helpful in solving problems with Diophantine equations, and a great deal of very deep and beautiful theory has been developed. However, a general theory that can deal with finding integral or rational solutions of arbitrary algebraic equations, or even with determining whether such solutions exist, is still a long way off.
In 1970 Yuri Matiyasevich gave a negative solution of the so-called 10th Hilbert problem by showing that there is no general decision procedure to determine whether solutions of an arbitrary Diophantine equation exist. Subsequently Matiyasevich showed that equations with as few as 9 variables lack a decision procedure. It is possible there may be equations with even fewer variables that lack such a procedure. However, the methods used in Wiles' proof of Fermat's Last Theorem indicate that all equations in 3 variables should be decidable. It is at present an open question what the least number of variables of an undecidable equation might be.
Nevertheless, we do have some very illuminating theory. For instance, there is a very celebrated conjecture stated by Louis Mordell in 1922 which says, roughly, that if P(x,y) is a polynomial with rational coefficients, then the equation P(x,y)=0 has only finitely many rational solutions (provided P(x,y) also has a "genus" greater than 1). This was an open problem until a proof was finally given by Gerd Faltings in 1983. The result can be interpreted geometrically as saying that a surface which contains all complex number solutions of the equation has only finitely many points on it with rational coordinates.
The most famous Diophantine equation of all, of course, is Fermat's equation: xn + yn = zn. Andrew Wiles finally proved in 1995 that, if n is 3 or more, this equation has no nonzero integral solutions -- even though Fermat had conjectured as much more than 350 years earlier. The final solution turned out to require extremely sophisticated tools from algebraic number theory, algebraic geometry (such as the theory of elliptic curves), and much more besides from a great deal of the remainder of modern mathematics.
Note that Fermat's problem is really about solutions of an infinite set of equations (one for each value of n). Technically speaking, if n were regarded as one of the variables in the equation, it would no longer be algebraic. Many other famous Diophantine problems have this feature, where one of the "variables" occurs as an exponent.
Elementary examples
Diophantine equations need not be limited to equations in only one variable (such as x2-2 = 0). It's frequently more interesting to consider equations with several variables, such as the Fermat equation. Just about the simplest equation with two variables and degree higher than 1 is x2-y2 = 0. (The degree of a single term is the sum of exponents of all variables occuring in the term. The degree of the equation is the largest of the degrees of its terms.) But this equation isn't very interesting, because it can be written as a product of polynomials of first degree: (x+y)(x-y) = 0. When the polynomial in such an equation is a product of factors of degree one having suitable coefficients, the solutions consist of the solutions of any of the factors, and those are trivial.
The next simplest second degree equation in two variables is x2+y2 = 1. (The equation x2+y2 = 0 has only the trivial solution x=y=0 when excluding complex numbers.) You probably recognize that as the equation of a circle of radius 1. Any point on the circle has coordinates satisfying the equation, and vice versa. The polynomial x2+y2-1 does not factor into first degree polynomials with real (i. e., not imaginary) coefficients. Nevertheless, it has solutions for x and y that are rational numbers, and even integers, such as (x,y) = (±1,0) or (0,±1). Clearly these are the only integer solutions, because we must have -1≤x≤1, so x has to be -1, 0, or 1, and so does y.
It was Diophantos himself who discovered all the rational solutions of this equation. He found what is called a parametric solution, obtained via elementary algebra. Namely, if t is any rational number (or even complex number, for that matter), then
x=(1-t2)/(1+t2), y=2t/(1+t2)
gives a point on the circle, and the coordinates are obviously rational if t is. Conversely, suppose (X,Y) is any point on the circle. It also lies on a straight line passing through the point (-1,0), having some slope t. The equation of this line is Y=t(X+1). The line intersects the circle at a unique point besides (-1,0), and if you do the algebra, you see that point must be given by X=(1-t2) and Y=2t/(1+t2). Since t = Y/(X+1), t must be rational if X and Y are. This shows that every rational point (except (-1, 0)) on the circle has the parametric form given, for some rational t. I. e., there's a very tidy 1:1 correspondence between rational numbers and points on the circle with rational coordinates.
This is actually just a special case of a second degree equation in three variables, which was extremely important to Greek mathematicians, namely x2+y2 = z2. Geometrically, this is the equation of a circle of radius z, centered on the origin. But more importantly to the Greeks, if x and y are the lengths of the sides of a right angled triangle, the length of the hypotenuse is z. This equation was discovered by Pythagoras (or someone in his school) some time in the 6th century BCE, and (of course) it was known as the Pythagorean theorem. If the lengths of the sides of the triangle are whole numbers, it is not necessarily true that the so is the length of the hypotenuse, since all we know is z=√(x2+y2). Indeed, the length isn't necessarily even rational – a fact which was considered quite scandalous at the time.
But if the length of the hypotenuse is a whole number, then (x,y,z) is called a Pythagorean triple. Such a triple was held in mystical regard by the Pythagoreans, so the search for such triples assumed a high importance. This search amounted to finding integer solutions of what we now call an instance of a Diophantine equation in 3 variables. Note that any solution in rational numbers also yields an integral solution. This is because if (x,y,z) is a rational triple, then multiplying through by the least common multiple of the denominators gives a triple of integers. (This works because the equation is homogeneous, with all terms having the same degree.)
No lesser a man than Euclid discovered how to describe all solutions in (positive) integers of this equation. Again, this is given in parametric form:
x=(u2-w2)w, y=2uvw, z=(u2+w2)w
It's trivial to check that you get a soution of the equation for any integer u, v, w (including 0 and negative values). The proof that this gives all solutions is harder and we leave that for you to think about.
Pell's equation
With the next step up in complexity, we reach some interesting territory that's directly connected with algebraic number theory. This involves an equation of the form x2 - ny2 = 1 for some positive integer n. n is fixed and not an additional variable. Solutions of this equation, however, depend very much on the arithmetic properties of n, and in some sense help us understand these properties.
The name "Pell's equation" was conferred by Leonhard Euler, who mistakenly gave credit for it to the otherwise obscure mathematician John Pell. However, the history of the equation goes all the way back to the Greeks, who were especially concerned with the case n=2, because it shines some light on the number √2. As you recall, √2 was of great interest (or concern) to them, ever since they realized √2 is not a rational number.
If y=0, then x=±1, so let's exclude that trivial solution. Then we can rewrite the equation as (x/y)2 = n+1/y2. Suppose there exist infinitely many solutions of the equation (as is in fact the case). Taking one with y large enough, 1/y2 can be arbitrarily small. And therefore, a solution (x,y) gives us a rational number x/y as close as we like to √n. This fact was already known implicitly to the Greeks, who also understood limit arguments. So they were pleased to know that even though √2 isn't rational, it can be approximated arbitrarily well by rational numbers. And the same is true of √n if n isn't a perfect square, so that √n is not rational.
Rather than pursue the details of Pell's equation, we need to look at how it's related to algebraic number theory. Recall that the set of all rational numbers is usually denoted by Q. Mathematically, this set has the structure known as a ring, in that it has the arithmetic operations of addition, subtraction, and multiplication, which observe certain rules. In fact, Q also has the structure of a field, which also allows for division by any of its elements other than 0. We will say much more about rings and fields in the next installment, so let's dispense with the formalities for now. It's safe, for present purposes, to think of the arithmetic operations of rings and fields as the ones you are already quite familiar with.
The integers, Z, also form a ring. Let Z[√n] stand for the set of all polynomials in powers of √n with coefficients in Z, and likewise for Q[√n] (with coefficients in Q). If n is a perfect square, these sets are just Z and Q, so let's assume n isn't a prefect square. It's obvious that these are rings also. But they are not necessarily fields, since 1/√n isn't rational if √n isn't.
What may be surprising, however, is that some elements of the form a+b√n may have inverses in these rings even if b≠0 and a≠±1. For instance, let n=2, and consider the number α=3+2√2 in Z[√2]. Observe that (3+2√2)(3-2√2) = 1. Hence 1/α = 1/(3+2√2) = 3-2√2, which is also an element of Z[√2]. Thus α has an inverse in Z[√2]. Such an element is called aunit. The only units of Z itself are ±1. But it turns out that for any positive n that's not a perfect square, the ring Z[√n] has infinitely many units.
What do these units look like? Well, first we need to define one more thing. If a+b√n is in Z[√n], define the function N(a+b√n) to be the number (a+b√n)(a-b√n) = a2-nb2. (This is the same definition as the square of the norm of a complex number, where n=-1.) Clearly, N(α) is also an integer for any α∈Z[√n]. It's simple algebra to show that if α,β∈Z[√n], then N(αβ)=N(α)N(β).
Given all this, if α∈Z[√n] is a unit, with inverse 1/α, so α(1/α)=1, we must have N(α)=1/N(1/α) is an integer, so N(α) is a unit of Z, which means it must be ±1. Conversely, if N(&alpha) = N(a+b√n) = (a+b√n)(a-b√n) = ±1, then α is a unit because it has an inverse. Hence the units α of Z[√n] are precisely the α such that N(α)=±1.
This is what allows us to show that Pell's equation has an infinite number of solutions, if there are any at all besides ±1 (which we haven't actually shown here). This is because every solution of Pell's equation a2-nb2 = 1 gives us a unit α = a+b√n with N(α)=1. For any such α, N(±αk)=1 too, for any k∈Z by the above, hence all ±αk are units, and give solutions of Pell's equation. Moreover, these are all distinct, because the only rational numbers that are units are ±1 (when b=0). For any irrational unit α the absolute value |α|≠1. We may assume |α|>1 (otherwise use 1/α). Then αk are distinct numbers for all positive k∈Z because the absolute values |αk|=|α|k are all distinct.
There may be other units as well, satisfying a2-nb2 = -1. For example, if n=5, we have N(2+√5)=-1, so 2+√5 is a unit of Z[√5]. If there exists a unit β = a+b√n with N(β)=-1, then there are also an infinite number of solutions of a2-nb2 = -1, corresponding to the units βk for odd integers k. (If k is even, we get solutions of Pell's equation.)
Catalan's conjecture
Catalan's conjecture can be stated as saying: the only consecutive whole numbers that have powers which differ by just one is the pair (2,3) -- where 32 - 23 = 1. Stated in the form of a Diophantine equation, the conjecture says there is only one solution of the equation xp - yq = 1, where x, y, p, and q are each more than 1, and x and y are consecutive (either y=x+1 or vice versa). Note that, like the Fermat problem, this is an assertion about an infinite set of possible equations.
To be sure, this problem sounds rather specialized, but it's still rather surprising to imagine that only one set of numbers meeting the stated conditions can exist. There are, after all, infinitely many pairs of consecutive numbers, and we ask only that there be two different powers of them which differ by one.
It's still an open question, but there has been recent progress. In 1976 it was shown that there can be at most finitely many exponents p, q for which any solution can be found. In other words, there is some (probably very large) number M for which all solutions have p and q less than M. Theoretical studies have discovered suitable values of M. (1017 will do.)
Then in 2000 Preda Mihailescu found that, except for 2 and 3, p and q must be what are called "double Wieferich primes", i. e. p(q-1) must be congruent to 1 modulo q2 and q(p-1) must be congruent to 1 modulo p2. Wieferich primes turn up in other number theoretical problems, but this is obviously a rather restrictive condition. (2 and 3 do not fit this criterion, but as often happens, very small primes are a special case.)
Given this result, Catalan's conjecture could be decided by computation alone, if all pairs of double Wieferich primes below the known limits can be identified. But a purely theoretical solution also seems tantalizingly close.
Beal's conjecture
Beal's conjecture is:
There are no positive integers x, m, y, n, z, r satisfying the equation
xm + yn = zr
where m, n, r > 2 and x, y, z are co-prime (have no prime factors in common)

facts about salts

facts about salts

ffaaccttss about salts:
1. Until the 20th century, a bag of salt (called amoleh) is the base currency in Abyssinia (now Ethiopia)

2. The Amazing Salar de Uyuni, the world's largest salt flat denan size of 4000 square miles in Bolivia. when a thin layer of sea water above the sand, it becomes a mirror rara avis. This reflectivity makes it a very useful tool in the calibration of scientific equipment from outer space. useful in the calibration of scientific equipment from outer space. The salt flat also contains half the world's supply of lithium.

3. Salt is very important for our bodies. if you drink too much water, it will eliminate a lot of fluid containing sodium in the body and can lead to hyponatremia (complications of medical disease in which the body loses a lot of fluids containing sodium) that prove fatal. This is what killed Jennifer Strange who follow the competition "Hold your wee for a wii"

4. even so important, consuming too much salt can also be deadly. You need about 1 gram of salt in every kilogram of your weight and that is enough to kill you. (So if you weigh 50 kg, 50 grams of salt can kill you, less LBH so) This is used as a method of ritual suicide in China, especially in the nobility as the time it is very expensive salt

5. Good quality sea salt contains many minerals that are important to our bodies. The best type of sea salt should be slightly damp when it was taken from the sea

6. In the Middle Ages, salt was so expensive that is sometimes referred to as "white gold". At that time, there are specific transportation routes for connecting the inland city of salt from Lüneburg to the German Baltic coast.

7. Black violations made in India by salt water mixed with the seeds of Harad. The mixture was allowed to evaporate leaving a black lump of salt. When the salt evenly, the resulting powder into the color pink

8. In Guerande, France, salt is still collected in the traditional way by using the basket which must wait for the sea water flooded. This enacting of salt is very expensive and highly sought after, especially with the best quality which is called the Fleur de cells (flower of salt)

9. there is a common misconception that the Roman soldiers paid with salt (as salary). but they are paid with money sabenarnya normal reply. connection with the salt is possible by the fact that the Roman soldiers are protected when traveling to Rome Salt

10. before biblical Judaism was removed, mixed with salt to sacrifice animals to God. according to Moses in Leviticus 2:13 which states: "Whatsoever sacrifice thou offerest, thou shalt season it with salt, neither shalt thou take away the salt of the covenant of thy God from thy sacrifice. In all thy oblations thou shalt offer salt. "(Fear of translating). So salt is considered a symbol of wisdom.

11. After the aviation fuel was purified, mixed with salt it to remove all traces of water before it can be used.

12. Sodium chloride is formed when sodium metal reacts with chlorine gas is stable. This is the only family of rocks on a regular basis are eaten by humans

13. In the early 1800s salt 4 times more expensive as the price of beef which are in the border region - it is very important in maintaining human and livestock in order to stay alive

14. Only 6% of salt used in the United States is used in food, 17% are used for de-icing streets and highways in winter

15. In the late 17th century, salt was the goods carried by the leading cargo from the Caribbean to North America and vice versa. It is used for food the slaves on sugar plantations

some facts

A kind of jellyfish (Turritopsis nutricula) upon reaching adulthood can transform itself back to childhood by converting its cells. It may repeat this to live forever.

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To avoid predators, a mother Slow Loris licks its offspring with poison before sending them off to search for food.

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The fungus Cordyceps is able to "mind-control" other insects like ants to climb plants and attach there to become its food.

why sun is hot

The Sun is hot, as the more astute of you will have noticed. It is hot because its enormous weight – about a billion billion billion tons – creates vast gravity, putting its core under colossal pressure. Just as a bicycle pump gets warm when you pump it, the pressure increases the temperature. Enormous pressure leads to enormous temperature. 

If, instead of hydrogen, you got a billion billion billion tons of bananas and hung it in space, it would create just as much pressure, and therefore just as high a temperature. So it would make very little difference to the heat whether you made the Sun out of hydrogen, or bananas, or patio furniture. 

Edit: this might be a little confusing. The heat caused by the internal pressure would be similar to that of our Sun. However, if it's not made of hydrogen, the fusion reaction that keeps it going wouldn't get under way: so a banana Sun would rapidly cool down from its initial heat rather than burning for billions of years. Thanks to people who pointed this out. 

All the matter that makes up the human race could fit in a sugar cube 

Atoms are 99.9999999999999 per cent empty space. As Tom Stoppard put it: "Make a fist, and if your fist is as big as the nucleus of an atom, then the atom is as big as St Paul's, and if it happens to be a hydrogen atom, then it has a single electron flitting about like a moth in an empty cathedral, now by the dome, now by the altar." 

If you forced all the atoms together, removing the space between them, crushing them down so the all those vast empty cathedrals were compressed into the first-sized nuclei, a single teaspoon or sugar cube of the resulting mass would weigh five billion tons; about ten times the weight of all the humans who are currently alive. 

Incidentally, that is exactly what has happened in a neutron star, the super-dense mass left over after a certain kind of supernova. 

Events in the future can affect what happened in the past 

The weirdness of the quantum world is well documented. The double slit experiment, showing that light behaves as both a wave and a particle, is odd enough – particularly when it is shown that observing it makes it one or the other. 

But it gets stranger. According to an experiment proposed by the physicist John Wheeler in 1978 and carried out by researchers in 2007, observing a particle now can change what happened to another one – in the past. 

According to the double slit experiment, if you observe which of two slits light passes through, you force it to behave like a particle. If you don’t, and observe where it lands on a screen behind the slits, it behaves like a wave. 

But if you wait for it to pass through the slit, and then observe which way it came through, it will retroactively force it to have passed through one or the other. In other words, causality is working backwards: the present is affecting the past. 

Of course in the lab this only has an effect over indescribably tiny fractions of a second. But Wheeler suggested that light from distant stars that has bent around a gravitational well in between could be observed in the same way: which could mean that observing something now and changing what happened thousands, or even millions, of years in the past. 

Almost all of the Universe is missing 

There are probably more than 100 billion galaxies in the cosmos. Each of those galaxies has between 10 million and a trillion stars in it. Our sun, a rather small and feeble star (a “yellow dwarf”, indeed), weighs around a billion billion billion tons, and most are much bigger. There is an awful lot of visible matter in the Universe. 

But it only accounts for about two per cent of its mass. 

We know there is more, because it has gravity. Despite the huge amount of visible matter, it is nowhere near enough to account for the gravitational pull we can see exerted on other galaxies. The other stuff is called “dark matter”, and there seems to be around six times as much as ordinary matter. 

To make matters even more confusing, the rest is something else called “dark energy”, which is needed to explain the apparent expansion of the Universe. Nobody knows what dark matter or dark energy is. 

Things can travel faster than light; and light doesn’t always travel very fast 

The speed of light in a vacuum is a constant: 300,000km a second. However, light does not always travel through a vacuum. In water, for example, photons travel at around three-quarters that speed. 

In nuclear reactors, some particles are forced up to very high speeds, often within a fraction of the speed of light. If they are passing through an insulating medium that slows light down, they can actually travel faster than the light around them. 

When this happens, they cause a blue glow, known as “Cherenkov radiation ”, which is (sort of) comparable to a sonic boom but with light. This is why nuclear reactors glow in the dark. 

Incidentally, the slowest light has ever been recorded travelling was 17 meters per second – about 38 miles an hour – through rubidium cooled to almost absolute zero, when it forms a strange state of matter called a Bose-Einstein condensate. 

Light has also been brought to a complete stop in the same fashion, but since that wasn't moving at all, we didn't feel we could describe that as "the slowest it has been recorded travelling". 

There are an infinite number of mes writing this, and an infinite number of yous reading it 

According to the current standard model of cosmology, the observable universe – containing all the billions of galaxies and trillions upon trillions of stars mentioned above – is just one of an infinite number of universes existing side-by-side, like soap bubbles in a foam. 

Because they are infinite, every possible history must have played out. But more than that, the number of possible histories is finite, because there have been a finite number of events with a finite number of outcomes. The number is huge, but it is finite. So this exact event, where this author writes these words and you read them, must have happened an infinite number of times. 

Even more amazingly, we can work out how far away our nearest doppelganger is. It is, to put it mildly, a large distance: 10 to the power of 10 to the power of 28 meters. That number, in case you were wondering, is one followed by 10 billion billion billion zeroes 

Black holes aren’t black 

They’re very dark, sure, but they aren’t black. They glow, slightly, giving off light across the whole spectrum, including visible light. 

This radiation is called “Hawking radiation”, after the former Lucasian Professor of Mathematics at Cambridge University Stephen Hawking, who first proposed its existence. Because they are constantly giving this off, and therefore losing mass, black holes will eventually evaporate altogether if they don’t have another source of mass to sustain them; for example interstellar gas or light. 

Smaller black holes are expected to emit radiation faster compared to their mass than larger ones, so if – as some theories predict – the Large Hadron Collider creates minuscule holes through particle collisions, they will evaporate almost immediately. Scientists would then be able to observe their decay through the radiation. 

The fundamental description of the universe does not account for a past, present or future 

According to the special theory of relativity, there is no such thing as a present, or a future, or a past. Time frames are relative: I have one, you have one, the third planet of Gliese 581 has one. Ours are similar because we are moving at similar speeds. 

If we were moving at very different speeds, we would find that one of us aged quicker than the other. Similarly, if one of us was closer than the other to a major gravity well like the Earth, we would age slower than someone who wasn’t. 

GPS satellites, of course, are both moving quickly and at significant distances from Earth. So their internal clocks show a different time to the receivers on the ground. A lot of computing power has to go into making your sat-nav work around the theory of special relativity. 

A particle here can affect one on the other side of the universe, instantaneously 

When an electron meets its antimatter twin, a positron, the two are annihilated in a tiny flash of energy. Two photons fly away from the blast. 

Subatomic particles like photons and quarks have a quality known as “spin”. It’s not that they’re really spinning – it’s not clear that would even mean anything at that level – but they behave as if they do. When two are created simultaneously the direction of their spin has to cancel each other out: one doing the opposite of the other. 

Due to the unpredictability of quantum behaviour, it is impossible to say in advance which will go “anticlockwise” and the other “clockwise”. More than that, until the spin of one is observed, they are both doing both. 

It gets weirder, however. When you do observe one, it will suddenly be going clockwise or anticlockwise. And whichever way it is going, its twin will start spinning the other way, instantly, even if it is on the other side of the universe. This has actually been shown to happen in experiment (albeit on the other side of a laboratory, not a universe). 

The faster you move, the heavier you get 

If you run really fast, you gain weight. Not permanently, or it would make a mockery of diet and exercise plans, but momentarily, and only a tiny amount. 

Light speed is the speed limit of the universe. So if something is travelling close to the speed of light, and you give it a push, it can’t go very much faster. But you’ve given it extra energy, and that energy has to go somewhere. 

Where it goes is mass. According to relativity, mass and energy are equivalent. So the more energy you put in, the greater the mass becomes. This is negligible at human speeds – Usain Bolt is not noticeably heavier when running than when still – but once you reach an appreciable fraction of the speed of light, your mass starts to increase rapidly.