[math-fun] What use are primes?
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look? Thanks, -- Stan -- Stan Isaacs 210 East Meadow Drive Palo Alto, CA 94306 stan@isaacs.com
Stan: Teach her about integer computations mod p, mod q, etc., and then the Chinese Remainder Theorem. She can use single precision integer math (except for the CRT part) to do matrix math _exactly_, without any roundoff error. Teach her some of the Babylonian & Egyptian diophantine equations. I got a summer course in number theory while a high school student (courtesy of the National Science Foundation) and we studied "Number Theory and its History" by Oystein Ore, and I fell in love. I had the same problem as she did, but I eventually got over it -- once you see how cool number theory is, who cares if it is useful or not! (Oystein Ore, Dover Publications, ISBN #0-486-65620-9 -- i.e., _cheap_, if you can find it.) If she is a really good student, whisper "Knuth" into her ear. His 3-part tome is the best source of cool math applied to useful things that has ever been written. It is expensive, it is very dense, and it will take her the rest of her professional life to grok it, but it will provide her with more interest & satisfaction than any 500 other books. See also: "The Unreasonable Effectiveness of Number Theory" http://www.amazon.com/gp/product/0821855018/ref=nosim/104-1699250-7817501?n=... At 05:13 PM 4/12/2006, Stan E. Isaacs wrote:
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look?
Thanks,
-- Stan -- Stan Isaacs 210 East Meadow Drive Palo Alto, CA 94306 stan@isaacs.com
At 08:13 PM 4/12/2006, Stan E. Isaacs wrote:
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers?
One thing is that there are several functions such that if you know their value on primes and powers of primes, you can easily compute the function on composite numbers. Stated another way, if you can factor a composite number into its prime factors, then it is easy to compute the function.
she asked what was the use of prime numbers? (besides being a fascination to math people)
Hardy and Koblitz say,
... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. -- G. H. Hardy, _A Mathematician's Apology_, 1940
G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). -- Neal Koblitz, _A Course in Number Theory and Cryptography_, 1987
To me, it looks like Hardy is mostly right, even now. Computer-based communication seems to be the sole killer-application for number theory. Will someone please disagree? -- Don Reble djr@nk.ca -- This message has been scanned for viruses and dangerous content by MailScanner, and is believed to be clean.
To me, it looks like Hardy is mostly right, even now. Computer-based communication seems to be the sole killer-application for number theory.
Will someone please disagree?
-- Don Reble djr@nk.ca
Don't primes come up all over in science? I recall there is a species of insect that has massive reproduction every 17 years to avoid competing with other species, and that it needed to be a prime number to avoid species with cycles in the 2 or 3 or 5 year ranges. Aren't primes big in things like crystallography? --Erik N
* Don Reble <djr@nk.ca> [Apr 13. 2006 08:10]:
she asked what was the use of prime numbers? (besides being a fascination to math people)
Hardy and Koblitz say,
... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. -- G. H. Hardy, _A Mathematician's Apology_, 1940
G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). -- Neal Koblitz, _A Course in Number Theory and Cryptography_, 1987
To me, it looks like Hardy is mostly right, even now. Computer-based communication seems to be the sole killer-application for number theory.
Will someone please disagree?
Me. Factorization into prime powers gives you the structure of the multiplicative group. Prerequisite to do basically anything. That we use computers instead of paper charts and hand computations makes no difference IMO. All FFT algorithms exploit the structure of the transform length. Rader's algorithm for prime length FFTs exploits the fact that a primitive root exists. Random number generators, and secure ones (BBS). Pseudo noise sequences via shift registers (for, e.g. spread spectrum). Watermarking with quadratic residues. Constructions for Hadamard matrices (shift registers, or, again quad.res.). Fast computations using CRT decomposition. [fill in 20 more here] Remove number theory from all electronic devices and nothing but flashlights would work a microsecond later. Just six month later there would probably be little left of what we consider a civilized world. Very very many people would die. I cannot see how the overwhelming importance of number theory (together with ECCs) should taint the underlying mathematics. I strongly dislike the attitude reflected by Hardy's words. Says jj, the computationalist P.S.: Gauss did a huge amount of computations.
Quoting "Stan E. Isaacs" <stan@isaacs.com>:
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. ....
I was going to mention the incommensurable breeding cycle of locusts but someone beat me to it. But how about turning the question around and asking about what good are non-primes? Suppose we had 23 hour days, or sold bread in baker's dozens, or (heaven forbid) tried to use the metric system. There wouldn't be any half-pints of beer, or packaging eggs in rectangular crates or doleing out pizza slices at a kids' birthday party. By knowing where the primes lie, we can avoid all this anguish. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
The context of the student's question made it seem like primes were "invented" in order to solve some particular problem. However, primes drop out naturally from the basic axioms of arithmetic (Peano), so they are there whether you want them or not. Since most people find the axioms of arithmetic pretty compelling, they get primes "for free". A crude analogy is the "two party" political system in the U.S. For better or for ill, the various constraints of the U.S. Constitution seem to force a two party solution. Given the number of challenges to this two party system over a long period of time, the continued dominance of the two parties indicates a remarkably stable equilibrium. The fact that this does not happen with other constitutional systems seems to indicate that some unique combination of features in the U.S. version conspire to produce this equilibrium. Of course, non-linear systems can have incredibly complex behaviors, with many different equilibrium points (both stable & non-stable), so we may yet see different behaviors from this system. Mathematicians analyze systems like these for their basic properties, while engineers try to take advantage of these properties in order to accomplish some particular goal. Mathematicians would continue to study primes forever, regardless of their "usefulness"; engineers would continue to attempt to "use" primes to build something. At 10:30 PM 4/12/2006, mcintosh@servidor.unam.mx wrote:
Quoting "Stan E. Isaacs" <stan@isaacs.com>:
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. ....
I was going to mention the incommensurable breeding cycle of locusts but someone beat me to it. But how about turning the question around and asking about what good are non-primes? Suppose we had 23 hour days, or sold bread in baker's dozens, or (heaven forbid) tried to use the metric system. There wouldn't be any half-pints of beer, or packaging eggs in rectangular crates or doleing out pizza slices at a kids' birthday party.
By knowing where the primes lie, we can avoid all this anguish.
- hvm
------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
There's also the matter of the free Riemann Gas, whose partition function is the Riemann zeta function. GW Mackey wrote this about it in 1978: G.W. Mackey, Unitary Group Representation in Physics, Probability, and Number Theory (Benjamin, 1978). "...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function -zeta'/zeta whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function." (p.300) Donald Spector, also unaware of Mackey's work, made a number of closely-related discoveries at the same time as Julia. D. Spector, "Supersymmetry and the Möbius inversion function", Communications in Mathematical Physics 127 (1990) 239. "We show that the Möbius inversion function of number theory can be interpreted as the operator (-1)F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that mu(n) = 0 when n is not squarefree being equivalent to the Pauli exclusion principle...One of the results we obtain is equivalent to the prime number theorem, one of the central achievements of number theory, in which the asymptotic density of prime numbers is computed." John Baez (my advisor) has this excellent article with plenty of references to more recent work: http://math.ucr.edu/home/baez/week199.html -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
PS. I stole most of this from Mark Watkins' pages on the zeta function: http://www.maths.ex.ac.uk/~mwatkins/zeta/physics.htm On 4/13/06, Mike Stay <mike@math.ucr.edu> wrote:
There's also the matter of the free Riemann Gas, whose partition function is the Riemann zeta function. GW Mackey wrote this about it in 1978:
G.W. Mackey, Unitary Group Representation in Physics, Probability, and Number Theory (Benjamin, 1978).
"...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function -zeta'/zeta whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function." (p.300)
Donald Spector, also unaware of Mackey's work, made a number of closely-related discoveries at the same time as Julia.
D. Spector, "Supersymmetry and the Möbius inversion function", Communications in Mathematical Physics 127 (1990) 239.
"We show that the Möbius inversion function of number theory can be interpreted as the operator (-1)F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that mu(n) = 0 when n is not squarefree being equivalent to the Pauli exclusion principle...One of the results we obtain is equivalent to the prime number theorem, one of the central achievements of number theory, in which the asymptotic density of prime numbers is computed."
John Baez (my advisor) has this excellent article with plenty of references to more recent work: http://math.ucr.edu/home/baez/week199.html -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
Stan E. Isaacs wrote
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look?
I suggest you take a look at the sixth section of Carl Friedrich Gauss' Disquisitiones Arithmeticae http://www.math.uni-bielefeld.de/~sieben/DA_sechster_abschnitt.pdf As an example Gauss calculates the decimal development of the fraction 6099380351/1271808720 (a fraction coming up as an approximation of \sqrt{23}) to as many digits as one wants by a method he invented to avoid long division. Prime numbers, `mods' and the `unique decomposition of a number into a product of primes' enter as efficient tools in a task that everyone knows from elementary school. Hereby Gauss' makes use of a table with the periods of the decimal fractions of the inverses of all primes and prime powers less than 1000: http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D137285 He finished to compute that table at the age of 18 -- the day before he left Brunswick to become a student in Goettingen. Best regards, Christian. -- Christian Siebeneicher e-mail:sieben@math.uni-bielefeld.de Universitaet Bielefeld http://www.math.uni-bielefeld.de/~sieben Fakultaet fuer Mathematik Postfach 10 01 31 tel secretary: (49) 0521-106 5022 D-33 501 Bielefeld fax: (49) 0521-106 6482 Germany tel home: (49) 0521-105335 fax home: (49) 0521-105325
Stan E. Isaacs wrote
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look?
I suggest you take a look at the sixth section of Carl Friedrich Gauss' Disquisitiones Arithmeticae http://www.math.uni-bielefeld.de/~sieben/DA_sechster_abschnitt.pdf As an example Gauss calculates the decimal development of the fraction 6099380351/1271808720 (a fraction coming up as an approximation of \sqrt{23}) to as many digits as one wants by a method he invented to avoid long division. Prime numbers, `mods' and the `unique decomposition of a number into a product of primes' enter as efficient tools in a task that everyone knows from elementary school. Hereby Gauss' makes use of a table with the periods of the decimal fractions of the inverses of all primes and prime powers less than 100 at the end of the Disquisitiones which is part of part of the much bigger table http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D137285 which he finished to compute the age of 18 -- the day before he left Brunswick to become a student in Goettingen. Best regards, Christian. -- Christian Siebeneicher e-mail:sieben@math.uni-bielefeld.de Universitaet Bielefeld http://www.math.uni-bielefeld.de/~sieben Fakultaet fuer Mathematik Postfach 10 01 31 tel secretary: (49) 0521-106 5022 D-33 501 Bielefeld fax: (49) 0521-106 6482 Germany tel home: (49) 0521-105335 fax home: (49) 0521-105325
participants (9)
-
Don Reble -
Erik Neumann -
Henry Baker -
Joerg Arndt -
Jud McCranie -
mcintosh＠servidor.unam.mx -
Mike Stay -
sieben＠math.uni-bielefeld.de -
Stan E. Isaacs