You will remember that a prime number is an integer that is evenly divisible only by 1 and itself. The numbers 2 and 3 are prime; 4 (which is divisible by 2) is not. The only even prime is 2, as all other even numbers are evenly divisible by 1, themselves, and 2. (The number 1 is usually not considered a prime, for technical reasons.)
Euclid (or one of his cronies) wondered whether there was a finite number of primes or an infinite number of them. In other words, with reference to all primes—2, 3, 5, 7, 11, 13, 17 … p—is there always a prime larger than p? Euclid’s proof is the quintessence of mathematical simplicity and beauty—in other words, elegance.
Let us assume that the primes are finite, so there is a largest prime. We will call that prime p. Multiply all the primes up to and including p (i.e., 2 × 3 × 5 × 7 … p). Call the product Q. (You could instead multiply all the integers—i.e., 1 × 2 × 3 × 4 × 5 × 6 × 7 … p—but this amounts to the same thing because the nonprime numbers reduce to primes.)
Now add 1 to Q.
Q + 1 is not evenly divisible by 2 or 3 or 5 or 7 … p or any combination of these (i.e., division by any of these leaves a remainder).
If Q + 1 is composite (i.e., not prime), it is evenly divisible by a prime larger than p (not p or any prime smaller than p because these leave a remainder). Therefore, p is not the largest prime.
If, on the other hand, Q + 1 is prime, then it is a prime larger than p.
In either case, there is a prime larger than p.
Should Q + 1 get cocky and think that it is the largest prime, we need merely give Q + 1 the role previously played by p and hire S to play Q’s role. (Ad infinitum.)
Consider this example. Say we make p = 3. Then Q = 2 × 3 = 6 and Q + 1 = 7, a prime number larger than 3.
If, on the other hand, we make p = 13 (i.e., 2 × 3 × 5 × 7 × 11 × 13), then Q = 30,030. And Q + 1 = 30,031, which is not prime and must require a prime larger than p.
Because 30,031 is a small number, it is known that its prime divisors are 59 and 509. You learn this from a table of prime numbers. For really large numbers, it takes centuries or much, much longer to determine their primes. That is what makes “unbreakable” codes unbreakable. In practice, if you had billions and billions or more of years, you could solve any practical code.
In other words, this works if, instead of 3 or 13, we made p equal, say, a skillion, skillion, skillion (or any other integer). The logic is the same. (Actually, there is no number “skillion.” It just means “any really, really big number,” as big as you want.)
If you try this, say, a trillion times, you will find that most of the 1 + (2 × 3 × 5 …) are composite, not prime.
These Euclidians—they were really, really good.
Picture a line containing the integers (… −3, −2, −1, 0, 1, 2, 3 …). Starting at zero, toss a coin. If it lands heads, go right; if tails, go left. Do this forever. Common sense says that you can choose a number and that, sooner or later, you will land on it. Common sense is correct; on a one-dimensional surface, a line, you will, having forever, return to your starting point.
Do the same thing on a plane, using a spin wheel divided into four equal sections, 1, 2, 3, and 4. Section 1 is left, 2 is right, 3 is forward, and 4 is backward. Common sense says that you can choose a number and that, sooner or later, you will land on it. Common sense is correct; on a one-dimensional surface, a line, you will, having forever, return to your starting point.
Now, do the same thing in three dimensions, say, three checkerboards on top of one another. Use a spin wheel divided into the eighteen possible moves (i.e., straight up, straight down, up and left, etc.; movement on the same plane is not permitted because it confuses things, but it does not change them in any relevant way).
Common sense says that, sooner or later, you will come back to where you started. Common sense is incorrect. There is nearly a two-thirds chance you will never land on your original spot.
I do not understand this either. But it is true.
A prime, you will remember, is an integer evenly divisible only by 1 and itself. It is obvious that 2 can be the only even prime (because all other even numbers are divisible by 1, themselves, and 2).
Now, aside from having a quintessentially ecumenical-sounding name, the Prussian mathematician and historian Christian Goldbach, in a letter to the great Euler, is noted for one thing: in 1742, he conjectured that every even number is the sum of two odd primes or, in the case of 2, 1 + 1 (e.g., 2 = 1 + 1; 4 = 2 + 2; 6 = 3 + 3; 8 = 5 + 3; 10 = 5 + 5; 12 = 7 + 5, … 98 = 19 + 79 …).1
Is Goldbach’s simple conjecture true? Two and a half centuries later, we still do not know. We do know that every even number is the sum of not more than 300,000 primes.2 So all that need be done is to shave the 300,000 down to 2. We know that there is some number above which any number is the sum of not more than four primes. The problem is that no one knows what that number is. It is not 37.
We also know that every even number through 100,000,000 is the sum of two primes, but there are a lot more than 100,000,000 even numbers larger than 100,000,000. Indeed, the even numbers are unending.
In science, a hundred million cases without an exception would justify the strongest belief a scientist can hold, one that is always tentative (the next empirical case may be an exception), but one that permits science to proceed.
In mathematics, all that counts is proof. Nothing less than a proof that every even number is the sum of two primes (or an exception that would refute the conjecture) counts for anything.
According to The Guinness Book of World Records, the largest number ever used in a mathematical proof was “Graham’s Number” This number required the invention of a new notation system because, even using towers of exponents, there would not be enough room in the universe to express the number. Graham’s Number arises in a certain problem in combinatorics, the branch of mathematics that tends to produce the largest numbers. It is believed that the solution to this problem is either “Graham’s Number” or 6.3
Let’s say you weigh yourself twice, once on a spring scale (like the one in your bathroom) and once on a balance scale (the kind where you put the object to be weighed on one side and an equal weight on the other). Both scales say you weigh 150 pounds. Now go to the equator and do the same thing. Then to the North Pole. Then down in a deep mine. Then in an airplane.
The balance scale will always tell you that you weigh 150 pounds (i.e., your weight equals that of the “150 pound weight”). The spring scale, however, will tell you that you weigh more than this at the equator and less at the other places.
Here’s why. The balance scale measures your mass, and this is the same, relative to the weight on the other side of the balance, at all places. The spring scale measures your weight, the pull of the earth on you (or, more accurately, the pull of you and the earth on each other; but the former is much greater, unless you have really been overdoing it at the dinner table).
Because the earth is slightly fatter at the equator than at the poles (as a result of “centrifugal force”), there is more of the earth pulling on you at the equator and less at the other places. When you are deep in the mine, for example, the portion of the earth above you is pulling in the other direction from the portion of the earth that is below you, reducing the pull downward.
About 13.8 billion years ago, a hole in an infinite eternal nothingness gave birth to a singularity, a possibly infinitely small universe that contained (in the form of energy) all of the energy and mass now contained in the universe it has become. Out of nothingness we can make a hole through which can be born all logic and all mathematics. From nothingness we can make the set containing nothing (i.e., the empty set), which we write “{ }” or “0” and call “zero.” We can then make the set “{ { } }.” We can call this set “1.” We can then make the set containing the set containing the set containing nothingness. We can call this 2 … .
Is it the same nothingness from the nothingness in which the universe was born? Did mathematics (including the extensive mathematics that seem to have nothing to do with the physical world) somehow stream out of the same hole as the universe? The obvious answer is no: one is a physical nothingness of a virtual vacuum, while the other is an intellectual construct. But, somehow, the obvious answer does not seem obviously correct. And we do know that all of nature follows the rules of mathematics and is explicable in mathematics, even if not all of mathematics has physical counterparts. Perhaps such mathematics is a blueprint for what the physical universe will become.
Or maybe mathematics is relevant to the physical universe only in the ways we already know about. Whatever the case, such things are fun to think about.
You may have wondered why a number raised to the zero power is one. You may have thought that a number raised to the zero power should be itself (because the number is not multiplied by anything, so it should just remain itself). But then you probably then remembered that a number raised to the first power is itself, and you would not want a number raised to the power 0 to equal the same number as the number raised to the power 1.
But why couldn’t we have a number raised to the power 0 be 0, just as a number multiplied by 0 is 0? It is true that this would not be terribly satisfying (because the number is not being multiplied by zero, but by itself zero times). But, as W. V. Quine points out, the strongest reason why we cannot permit a number raised to the 0 power to be 0 is this:
We want n m + 1 always to be nm times n (for these are the same thing). That is, 23 + 1 = 23 × 2.
If we take m as 0, we get
n1 = n0 times n, which is the same as
n = n0 times n.
So n0 must be 1.
According to David Wells’s The Penguin Dictionary of Curious and Interesting Numbers, 39 is the first uninteresting (natural) number—which, of course, makes it interesting.
Hey, this is easy. 1, 2, 4, umm.
Wells also points out that nn + 1 is equal to a prime number when n is 1, 2, or 4.
11 + 1 = 2
22 + 1 = 5
44 + 1 = 257
If there is another prime that is generated by this method, it has, at minimum, three hundred thousand digits.
Judy is thirty-three, unmarried, and quite assertive. A magna cum laude graduate, she majored in political science in college and was deeply involved in campus social affairs, especially in antidiscrimination and antinuclear issues. Which statement is more probable?
(a)Judy works as a bank teller.
(b)Judy is active in the feminist movement and works as a bank teller.
Most people, correctly perceiving that Judy may well be a feminist, choose (b). But note that if (b) is true, then (a) must also be true. If Judy is a bank teller and a feminist, Judy must be a bank teller. There is, however, a possibility that (a) is true and (b) is not. That is, Judy is a bank teller and not a feminist. Thus, (a) is more probable.
This is a wonderful example, created by psychologists Amos Tversky and Daniel Kahneman, of the way in which people come to incorrect conclusions because they substitute inappropriate mental models for logical thinking.4
Here is a quotation whose source I have long since lost:
All that exists or could have existed or could come to exist—in the mind or in potential or in reality—is the set of all sets, which has the same structure as the set of all complex one-dimensional subspaces of a complex infinite-dimensional Hilbert space.
I have only the foggiest idea what this means, but it sure sounds nifty.
1.For simplicity’s sake, this is the way Goldbach’s conjecture is almost invariably presented. Apostolos Doxiadis, in his very fine novel Uncle Petros and Goldbach’s Conjecture has a footnote giving a more accurate description:
In fact, Christian Goldbach’s letter (to Euler) in 1742 contains the conjecture that “every integer can be expressed as the sum of three primes.” However, as (if this is true) one of the three such primes expressing even numbers will be 2 (the addition of three odd primes would be of necessity odd, and 2 is the only even prime number), it is an obvious corollary that every even number is the sum of two primes. Ironically, it was not Goldbach but Euler who phrased the conjecture that bears the other’s name—a little known fact, even among mathematicians.
2.We also know that every odd number is the sum of not more than 300,000 primes because we need merely add 1 (here considered a prime) to the primes that sum to an even number to get the primes that sum to next odd number.
3.David G. Wells, The Penguin Dictionary of Curious and Interesting Numbers, 1998.
4.From John Allen Paulos, Once upon a Number: The Hidden Mathematical Logic of Stories.