Prime number/Citable Version: Difference between revisions
imported>Greg Woodhouse (Euler's Proof - filled in) |
imported>Fredrik Johansson m (indent formulas) |
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Every integer ''N'' > 1 can be written in a unique way as a product of prime factors, up to reordering. to see why this is true, assume that ''N'' can be written as a product of prime factors in two ways | Every integer ''N'' > 1 can be written in a unique way as a product of prime factors, up to reordering. to see why this is true, assume that ''N'' can be written as a product of prime factors in two ways | ||
<math>N = p_1 p_2 \cdots p_m = q_1 q_2 \cdots q_n</math> | :<math>N = p_1 p_2 \cdots p_m = q_1 q_2 \cdots q_n</math> | ||
We may now use a technique known as [[mathematical induction]] to show that the two prime decompositions are really the ''same''. | We may now use a technique known as [[mathematical induction]] to show that the two prime decompositions are really the ''same''. | ||
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Given any prime divisor of ''N'' (call it ''p''), we know that | Given any prime divisor of ''N'' (call it ''p''), we know that | ||
<math>p | p_1 p_2 \cdots p_m</math> | :<math>p | p_1 p_2 \cdots p_m</math> | ||
and | and | ||
<math>p | q_1 q_2 \cdots q_n</math> | :<math>p | q_1 q_2 \cdots q_n</math> | ||
Because ''p'' is prime, we know there are integers ''i'' and ''j'' such that <math>p | p_i</math> and <math>p | q_j</math>. On the other hand, since <math>p_i</math> and <math>q_j</math> are prime, it must be that p is ''equal'' to <math>p_i</math> and to <math>q_j</math>. This means that if we set <math>N_1 = N/p</math>, we may write | Because ''p'' is prime, we know there are integers ''i'' and ''j'' such that <math>p | p_i</math> and <math>p | q_j</math>. On the other hand, since <math>p_i</math> and <math>q_j</math> are prime, it must be that p is ''equal'' to <math>p_i</math> and to <math>q_j</math>. This means that if we set <math>N_1 = N/p</math>, we may write | ||
<math>N_1 = p_1 p_2 \cdots \hat{p_i} \cdots p_m</math> | :<math>N_1 = p_1 p_2 \cdots \hat{p_i} \cdots p_m</math> | ||
and | and | ||
<math>N_1 = q_1 q_2 \cdots \hat{q_j} \cdots q_n</math> | :<math>N_1 = q_1 q_2 \cdots \hat{q_j} \cdots q_n</math> | ||
where the circumflex ("hat symbol") indicates that <math>p = p_i = q_j</math> is ''not'' part of the prime factorization of <math>N_1</math> | where the circumflex ("hat symbol") indicates that <math>p = p_i = q_j</math> is ''not'' part of the prime factorization of <math>N_1</math> | ||
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Suppose the set of prime numbers is finite, say <math>\{ p_1, p_2, p_3, \ldots, p_n \}</math>, and let | Suppose the set of prime numbers is finite, say <math>\{ p_1, p_2, p_3, \ldots, p_n \}</math>, and let | ||
<math>N = p_1 p_2 \cdots p_n +1</math> | :<math>N = p_1 p_2 \cdots p_n +1</math> | ||
then for each <math>i \in 1, \ldots, n</math> we know that <math>p_i \not| N</math> (because the remainder is 1). This means that ''N'' is not divisible by an prime, which is impossible. This contradiction shows that our assumption that there must only be a finite number of primes must have been wrong and thus proves the theorem. | then for each <math>i \in 1, \ldots, n</math> we know that <math>p_i \not| N</math> (because the remainder is 1). This means that ''N'' is not divisible by an prime, which is impossible. This contradiction shows that our assumption that there must only be a finite number of primes must have been wrong and thus proves the theorem. | ||
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The Swiss mathematician [[Leonhard Euler]] showed how the existence of infinitely many primes could be demostrated using a rather different approach. The starting point is the fact that the [[harmonic series]] | The Swiss mathematician [[Leonhard Euler]] showed how the existence of infinitely many primes could be demostrated using a rather different approach. The starting point is the fact that the [[harmonic series]] | ||
<math>1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + \ldots</math> | :<math>1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + \ldots</math> | ||
''diverges''. That is, for any <math>N > 0</math>, we can choose n such that | ''diverges''. That is, for any <math>N > 0</math>, we can choose n such that | ||
<math>1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > N</math> | :<math>1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > N</math> | ||
A second fact we will need about infinite series is that if <math>|x| < 1</math> | A second fact we will need about infinite series is that if <math>|x| < 1</math> | ||
<math>1 + \frac{1}{x} + \frac{1}{x^2} + \ldots = \frac{1}{1 - x}</math> | :<math>1 + \frac{1}{x} + \frac{1}{x^2} + \ldots = \frac{1}{1 - x}</math> | ||
Now, unique factorization gives us | Now, unique factorization gives us | ||
<math>\sum_{i = 1}^{\infty} \frac{1}{i} = \prod_p \frac{1}{1 - p}</math> | :<math>\sum_{i = 1}^{\infty} \frac{1}{i} = \prod_p \frac{1}{1 - p}</math> | ||
where the sum on the left is the harmonic series, and the product on the right is extended over all primes ''p''. Now, if there are only finitely many primes, the product on the right has a (finite) value, but the harmonic series (the sum on the left) ''diverges''. This shows that there must be infinitely many primes, after all. | where the sum on the left is the harmonic series, and the product on the right is extended over all primes ''p''. Now, if there are only finitely many primes, the product on the right has a (finite) value, but the harmonic series (the sum on the left) ''diverges''. This shows that there must be infinitely many primes, after all. | ||
Revision as of 07:38, 6 April 2007
A prime number is a whole number (i.e., one having no fractional or decimal part) that cannot be evenly divided by any numbers but 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. With the exception of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} , all prime numbers are odd numbers, but not every odd number is prime. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9 = 3\cdot3} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15 = 3\cdot5} , so neither 9 nor 15 is prime. The study of prime numbers has a long history, going back to ancient times, and it remains an active part of number theory (a branch of mathematics) today. It is commonly believed that the study of prime numbers is an interesting, but not terribly useful, area of mathematical research. While this used to be the case, the theory of prime numbers has important applications now. Understanding properties of prime numbers and their generalizations is essential to modern cryptography, and to public key ciphers that are crucial to Internet commerce, wireless networks and, of course, military applications. Less well known is that other computer algorithms also depend on properties of prime numbers.
Definition
Prime numbers are usually defined to be positive integers (other than 1) with the property that they are only (evenly) divisible by 1 and themselves. In other words, a number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} is said to be prime if there are exactly two Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m | n} , namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n} .
There is another way of defining prime numbers, and that is that a number is prime if whenever it divides the product of two numbers, it must divide one of those numbers. A nonexample (if you will) is that 4 divides 12, but 4 does not divide 2 and 4 does not divide 6 even though 12 is 2 times 6. This means that 4 is not a prime number. We may express this second possible definition in symbols (a phrase commonly used to mean "in mathematical notation") as follows: A number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in \mathbb{N}}
is prime if for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in \mathbb{N}}
such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | ab}
(read p divides ab), either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | a}
or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | b}
. If the first characterization of prime numbers is taken as the definition, the second is derived from it as a theorem, and vice versa.
- Aside on mathematical notation: The second sentence above is a translation of the first into mathematical notation. It may seem difficult at first (perhaps even a form of obfuscation!), but mathematics relies on precise reasoning, and mathematical notation has proved to be a valuable, if not indispensible, aid to the study of mathematics. It is commonly noted while ancient Greek mathematicians hd a good understanding of prime numbers, and indeed Euclid was able to show that there are infinitely many prime numbers, the study of prime numbers (and algebra in general) was hampered by the lack of a good notation, and this is one reason ancient Greek mathematics (or mathematicians) excelled in geometry, making comparatively less progress in algebra and number theory.
Unique Factorization
Every integer N > 1 can be written in a unique way as a product of prime factors, up to reordering. to see why this is true, assume that N can be written as a product of prime factors in two ways
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = p_1 p_2 \cdots p_m = q_1 q_2 \cdots q_n}
We may now use a technique known as mathematical induction to show that the two prime decompositions are really the same.
Given any prime divisor of N (call it p), we know that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | p_1 p_2 \cdots p_m}
and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | q_1 q_2 \cdots q_n}
Because p is prime, we know there are integers i and j such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | p_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p | q_j} . On the other hand, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j} are prime, it must be that p is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} and to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j} . This means that if we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = N/p} , we may write
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = p_1 p_2 \cdots \hat{p_i} \cdots p_m}
and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = q_1 q_2 \cdots \hat{q_j} \cdots q_n}
where the circumflex ("hat symbol") indicates that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_i = q_j} is not part of the prime factorization of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1}
Continuing this way, we obtain a sequence of numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = N_0 > N_1 > N_2 > \ldots > N_n = 1} where each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{\alpha}} is obtained by dividing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{\alpha - 1}} by a prime factor. In particular, we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n} and that there is some permutation ("rearrangement") of the indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1, 2, \ldots n} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i = q_{\sigma(i)}} . Said differently, the two factorizations of N must be the same up to a possible rearrangment of terms.
There are infinitely many primes
One basic fact about the prime numbers is that there are infinitely man of them. In other words, the list of prime numbers 2, 3, 5, 7, 11, 13, 17, ... doesn't ever stop. There are a number of ways of showing that this is so, but one of the oldest and most familiar proofs goes back go Euclid.
Euclid's Proof
Suppose the set of prime numbers is finite, say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ p_1, p_2, p_3, \ldots, p_n \}} , and let
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = p_1 p_2 \cdots p_n +1}
then for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in 1, \ldots, n} we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i \not| N} (because the remainder is 1). This means that N is not divisible by an prime, which is impossible. This contradiction shows that our assumption that there must only be a finite number of primes must have been wrong and thus proves the theorem.
Euler's Proof
The Swiss mathematician Leonhard Euler showed how the existence of infinitely many primes could be demostrated using a rather different approach. The starting point is the fact that the harmonic series
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + \ldots}
diverges. That is, for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N > 0} , we can choose n such that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > N}
A second fact we will need about infinite series is that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| < 1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \frac{1}{x} + \frac{1}{x^2} + \ldots = \frac{1}{1 - x}}
Now, unique factorization gives us
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i = 1}^{\infty} \frac{1}{i} = \prod_p \frac{1}{1 - p}}
where the sum on the left is the harmonic series, and the product on the right is extended over all primes p. Now, if there are only finitely many primes, the product on the right has a (finite) value, but the harmonic series (the sum on the left) diverges. This shows that there must be infinitely many primes, after all.
- Remark: One might well wonder what point there is in offering multiple proofs of the same result. After all, isn't one enough? In mathematics, the point of writing a proof is not so much to establish that something is true, but to understand why it is true. Euclid's proof is purely algebraic, and ultimately depends on the fact that prime numbers can be characterized in two different ways. Euler's proof, on the other hand, makes use of a couple of facts about infinite series combined with the unique factorization property establishe above. What is interesting is that these two proofs (and there are many others) use ideas from very different parts of mathematics to arrive at the same result. One has the feeling that this is evidence of particularly deep interconnections between different parts of mathematics.