Prime Number Formulas: Find the Nth Prime

[hadi amiri 4]

Is any formula that produces nth prime ?
if not visit this web site
http://www.primenumbersformula.com

 

[Santa1]

Doesnt m=10 give 21=3*7 ??

 

[morphism]

According to Maple, H(10)=2.

If you look at the formula closely, you'll notice that H(m) is going to be an odd prime whenever 2m+1 is a prime, and otherwise it's going to be 2. This follows from Wilson's theorem (2m+1 is a prime iff (2m)! + 1 = 0 (mod 2m+1)) and the behaviour of \left\lfloor\lfloor x \rfloor / x \right\rfloor. It's really nothing special.

 

[hadi amiri 4]

Please work on this formula and find all gaps in order to fin a formula for primes

 

[Hurkyl]

According to Maple, H(10)=2.

If you look at the formula closely, you'll notice that H(m) is going to be an odd prime whenever 2m+1 is a prime, and otherwise it's going to be 2. This follows from Wilson's theorem (2m+1 is a prime iff (2m)! + 1 = 0 (mod 2m+1)) and the behaviour of \left\lfloor\lfloor x \rfloor / x \right\rfloor. It's really nothing special.

Ah, so it's just a clever way to obscure a definition by cases, and is really just saying

H(m) = \begin{cases}<br /> 2m + 1 &amp; 2m + 1 \textbox{\ is\ prime} \\<br /> 2 &amp; \textbox{otherwise}<br /> \end{cases}

 

[Santa1]

According to Maple, H(10)=2.

I guess you're right, I must've had an error somewhere.

 

[HallsofIvy]

According to Maple, H(10)=2.

If you look at the formula closely, you'll notice that H(m) is going to be an odd prime whenever 2m+1 is a prime, and otherwise it's going to be 2. This follows from Wilson's theorem (2m+1 is a prime iff (2m)! + 1 = 0 (mod 2m+1)) and the behaviour of \left\lfloor\lfloor x \rfloor / x \right\rfloor. It's really nothing special.

Are you saying then that one can test whether an odd number is prime by calculating this? Wouldn't that be faster than, say, trial division by primes less than its square root?

 

[huba]

As long as (2m)! can be calculated (although it should be difficult for larger values of m), there is no need for the proposed formula, because Wilson's theorem already tests primality: if 2m+1 divides (2m)!+1 then 2m+1 is prime.
When (2m)! is difficult to calculate then the formula becomes difficult as well.

 

[huba]

Please work on this formula and find all gaps in order to fin a formula for primes

Using Hurkyl's definition for H(m),

H(m)=2 if 2m+1 is composite.
H(4) = 2, and for all k, H(4+3k)=2.
H(12)=2, and for all k, H(12+5k)=2,
H(24)=2, and for all k, H(24+7k)=2, etc.

This is Eratosthenes' sieve, and I know about one exact formula for primes which utilizes it: Riemann's solution for the prime counting function. What other formula can be found?

 

[Hurkyl]

Using Hurkyl's definition for H(m),

H(4) = 2, and for all k, H(4+3k)=2.
H(12)=2, and for all k, H(12+5k)=2,
H(24)=2, and for all k, H(24+7k)=2, etc.

Why would you think any of those claims are true? Did you actually try to evaluate them for any value of k?

This is Eratosthenes' sieve,

Sure doesn't look like it.

 

[hadi amiri 4]

The range of H is
{3,5,7,2,11,13,2,17,19,...}={2,3,5,...}
so there is no problem.am I right?

 

[huba]

Why would you think any of those claims are true? Did you actually try to evaluate them for any value of k?

H(m) = 2 if 2m+1 is composite.

When m = 4 + 3k, then 2m + 1 = 3(2k + 3) which is composite for all k. In fact, 3(2k + 3) gives all odd numbers > 9 that are divisible by 3.

Similarly for m = (p^2-1)/2 + pk where p is prime, because then 2m + 1 = p(2k + p), and so 2m+1 is composite, and so H(m) = 2.

 

[Hurkyl]

Good call; I had the wrong definition of H in my head when I responded.

 

[huba]

The range of H is
{3,5,7,2,11,13,2,17,19,...}={2,3,5,...}
so there is no problem.am I right?

The formula is interesting (it has certainly attracted many responses in a short time), but it does not produce the nth prime as effectively suggested in the first entry in this thread.
For a given n, we don't know for what value of m we get H(m) = the nth prime, until finding it by trial and error by calculating H(m) for a certain range of m. For instance, if the first prime is 2 (n=1), we find it (first) when m=4.

 

[CRGreathouse]

Are you saying then that one can test whether an odd number is prime by calculating this? Wouldn't that be faster than, say, trial division by primes less than its square root?

Trial division on n takes O(\sqrt n) arithmetic operations, or O(\sqrt n/\log n) with precomputation of primes. With Newton's method and Karatsuba, this takes O(n^{0.8}) bit operations. Without pecomputation, this method takes only logarithmic space -- log n space for n, log n space for the calculation, and 1/2 log n space for a counter.

A quick way to calculate \lfloor(2m)!/(2m+1)\rfloor is noting that if 2n+1 is composite, \lfloor(2m)!/(2m+1)\rfloor=(2m)!/(2m+1). But this is what we're trying to determine, so slower methods must prevail. Now there are methods for computing n! in O\left(n(\log n)^{2+\varepsilon}\right) bit operations, but these require factorization of n -- again, what we want to avoid. So binary splitting gives the solution in time O(n^{1.585}), though with lots of logarithmic factors hidden in the big O.* After that the exponentiation must be performed, at cost O(\log n!)=O(n\log n). The space requirement for a literal computation is huge: [log]O(\log (2n)!)=O(n\log n)[/tex], an order of magnitude larger than trial division.

In conclusion, trial division is far superior, taking roughly the square root of the time and the logarithm of the space of this method.

* At several points I'm able to hide logarithmic factors between lg 3 = 1.58496... and the 1.585 I display; sneaky, eh?

 

[majesticman]

guys if you all were able to generate a formula that gives the nth prime...you will be rich!...correct me if i am wrong but aren't security for top secret files as well as e-commerce based on the fact that really big prime numbers (they are set as encryption keys) cannot be factorized. So good luck...i thnk it is RSA algorithm that relies on prime numbers

 

[patrcik]

guys if you all were able to generate a formula that gives the nth prime...you will be rich!...correct me if i am wrong but aren't security for top secret files as well as e-commerce based on the fact that really big prime numbers (they are set as encryption keys) cannot be factorized. So good luck...i thnk it is RSA algorithm that relies on prime numbers

While I doubt you would get rich(If you were thinking about the RSA factorization prizes, I'm sorry to tell you, but they stopped giving them out in 2007), but you would secure your place in history.
Yes, the RSA uses prime numbers to encrypt. SSL uses prime numbers as well. If some smart guy(or gal) figures out the distribution of prime numbers those would be dropped pretty quickly.

 

[Hurkyl]

guys if you all were able to generate a formula that gives the nth prime...you will be rich!

There already exist explicit formulas for the n-th prime, and a variety of algorithms for computing the n-th prime.

correct me if i am wrong but aren't security for top secret files as well as e-commerce based on the fact that really big prime numbers (they are set as encryption keys) cannot be factorized..

I should hope not -- I can factor such numbers in my head, no matter how big they are.

 

[Hurkyl]

figures out the distribution of prime numbers

It's already known. The primes occur precisely when we have an integer that is not a multiple of smaller integers. Aside from that obvious fact, there are bewildering array of bulk statistics known about the primes, and a great many more that are consequences of the Riemann hypothesis.

 

[Kurret]

Why is the mathematical world so focused on prime numbers, why not the general case, to find all numbers with n prime factors?

 

[CRGreathouse]

guys if you all were able to generate a formula that gives the nth prime...you will be rich!...correct me if i am wrong but aren't security for top secret files as well as e-commerce based on the fact that really big prime numbers (they are set as encryption keys) cannot be factorized. So good luck...i thnk it is RSA algorithm that relies on prime numbers

1. There are lots of formulas that generate primes; creating a new one won't guarantee that you can get a paper published, much less than make you rich.
2. Many encryption algorithms rely on the hardness of factoring large semiprimes; factoring primes is, of course, trivial.

 

[CRGreathouse]

Why is the mathematical world so focused on prime numbers, why not the general case, to find all numbers with n prime factors?

I don't think there's any push to find (general) prime numbers -- there are too many -- but to learn how to quickly factor a number into its prime factors. This is interesting because such a factorization is unique (up to order and units), though there is still interest in factoring into coprimes and partial factorization in general.

 

[DeaconJohn]

Why is the mathematical world so focused on prime numbers, why not the general case, to find all numbers with n prime factors?

I don't think there's any push to find (general) prime numbers -- there are too many -- but to learn how to quickly factor a number into its prime factors. This is interesting because such a factorization is unique (up to order and units), though there is still interest in factoring into coprimes and partial factorization in general.

That's a good question, Kurret (IMHO), and that's a good reply, Greathouse (IMHO).

Kurret,

A few more thoughts, adding to what Greathouse said.

One reason there is a "focus" on "prime numbers" these days is the relation with the Riemann hypothesis. For example, if we can show that the error term in the prime number theorem grows at the order of the square root of n, then we will know that the Riemann Hypothesis is true. (ref: Barry Mazur's "error term" article in this issue of the BAMS; the equivalence was shown about 100 years ago).

So, this fact reduces your question to "why is there so much interest in the Riemann hypothesis"?

Well, first of all, the RH is a very old problem.
Second, if it is true, many intereesting and beautiful facts about nature will follow. (My viewpoint is that interesting mathematics describes interesting things about nature.)
Third, everything that is known about the RH gives the impression that we a re very close to solving it.
Fourth, it's been that way for about 100 years now, so why haven't we solved it? (Riemann even wrote down an exact formula for the error term in the prime number theorem. It would seem that all we have to do is analyze that formula.)
Fifth, numerical studies of the behavour of the zeroes of the zeta function on the critical line indicate that current techniques are inadequate to solve it. The behaviour of those zeroes is just too strange and too sbutle to get a handle on. Therefore, a solution would likely result in a significant advance in our understanding of mathematics in general.
Sixth, there is a million dollar prize for the person who resolves it.
Seventh, the name of the person who solves it is very likely to go down in history for a very long time.

On the other hand, as Greathouse pointed out, the question you ask is really of very high interest to today's mathematicians. One reason for that is IF we could find an effectgive algorithm to answer your question, THEN we would be a lot closer to finding an effective algorithm to factoring the product of two large primes in a short period of time. This would break RSA cryptography. It is highly likely that the proof would carry over to break discrete logarithm cryptography. It is quite possible that the proof would allow us to break elliptic curve cryptography.

The underlying mathematical meta-principle that you are touching on is that it is a lot easier to ask hard questions than it isw to ask "good" quesitons. A "good" question is a question that we have a hope of solving; that we feel that if we worked on it, we should be able to come up with something.

A small group of very good mathematicians are currently investigating the properties of numbers that are the product of a large number of small primes. The fact that that investigation is difficult (but doable) underlines just how hard the question that you ask probably is.

Another good example of a "hard" question is the question of whether or not there are an infinite number of Mersenne primes. For that one, there even exists a heuristic argument that there are an infinite number and it yields an asympototic estimate for how many there are!

And yet, as far as I know, nobody has any idea about how to get started on actually proving that there are an infinite number of them. Richard Guy also said this in his book "Open Problems in Number Theory" or something like that.

Oh, oh, oh, except the female star of the science fiction movie called "Proof." The "science fiction" in the movie is that she resolved the question about the finiteness of the Mersenne primes.

DJ

P.S. I welcome corrections, or even "that doesn't sound right" comments. That's how I learn.

 

[hadi amiri 4]

I have good news the discoverer of this formula is going to publish a book that contains many results like defining the set of twin prime numbers a formula about producing nth prime and many other top results .

 

[CRGreathouse]

I have good news the discoverer of this formula is going to publish a book that contains many results like defining the set of twin prime numbers a formula about producing nth prime and many other top results .

Well twin primes are old, at least as old as Sophie Germain (early 1800s) and probably older. I don't know hold long we've had explicit formulas for generating the primes, rather than just algorithms, but surely more than a hundred years.

What, then, are these "other top results"?

 

[DeaconJohn]

I have good news the discoverer of this formula is going to publish a book that contains many results like defining the set of twin prime numbers a formula about producing nth prime and many other top results .

You're kidding, right?

 

[hadi amiri 4]

why you don not respect others attempt
have you seen this book
why you are so judgemental

 

[Hurkyl]

why you don not respect others attempt
have you seen this book
why you are so judgemental

If you don't want people evaluating your hype, then don't post it. What else are we supposed to do with it?

 

[huba]

There is certainly no point in making qualifying statements about a claim that has not been presented in the necessary detail.

 

[CRGreathouse]

why you don not respect others attempt
have you seen this book
why you are so judgemental

We haven't seen the book, that's why I asked what the "other top results" were. It's hard to respect results you don't know about.

 

[thompson03]

I assume you guys are being sarcastic right?

There is no formula (function) which, given an arbitrary input will guarantee a prime output, let alone, give you the primes in order.

You can work recursively, (ie, the sieve of Eratosthanes) but its not really a formula, it just knocks out the composites one prime factor at a time.

Nor has it been proven (conclusively) that there are an infinitude of twin primes.

 

[CRGreathouse]

I assume you guys are being sarcastic right?

If you mean my post #25, I was not being sarcastic at all. If not, which posts were you referring to?

There is no formula (function) which, given an arbitrary input will guarantee a prime output, let alone, give you the primes in order.

MathWorld cites Ruiz 2000:
Ruiz, S. M. "The General Term of the Prime Number Sequence and the Smarandache Prime Function." Smarandache Notions J. 11, 59-61, 2000.

p_n=1+\sum_{k=1}^{2(\lfloor n\ln n\rfloor+1)}\left[1-\left\lfloor\sum_{j=2}^k1+\lfloor s(j)\rfloor}/n\right\rfloor\right]

where
s(j)=-\frac{\sum_{s=1}^j\left(\lfloor j/s\rfloor-\lfloor(j-1)/s\rfloor\right)-2}{j}There are many more examples, some simpler and some more complex. Some give all primes in order, some give all distinct primes, some give the count of primes through n, etc.

Nor has it been proven (conclusively) that there are an infinitude of twin primes.

No one on this thread has made that claim.

 

[thompson03]

f(x) = x ; if x is prime does too

let me rephrase,

there is no function that is not based on recursion or more fundamentally on a sieve process.

Making a function that step by step knocks out the composites is not exactly what I meant.

"My bad" on the linguistics there, sorry

 

[CRGreathouse]

let me rephrase,

there is no function that is not based on recursion or more fundamentally on a sieve process.

I suppose then that you'd consider Wilson's Theorem to be essentially a sieve?

 

[thompson03]

I'm not trying to be stupid, i really think this is a semantic / linguistic confusion, not an argument.

Looks to me like wilsons theorem is a test, not a function.


But, anything with factorials, list all previous values (factors).
So the Smarandache Prime Function essential shows that if you list out all of the numbers multiplied together, then there are sufficient factors to generate every number up to the current number, so if its prime, then the test holds.

straight from wolframs site:
/quote
For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!=4·3·2·1=8·3, so mu(8)=4
/endquote

So mu(5) will be 5 because 5 is prime, because none of the previous factors give multiples of five,
this is essentially saying f(x) = x is prime, if x is prime

All I mean is, there is no function F(M)=P for some input M that guarentees P is prime, and M is any single valued input, be it real, integer, complex or quaternian. If there where, it would be bloody famous and there'd be no need for prime testing algorithms. Especially if it was an invertable function!

But think of it, if you had such a function and you wanted to test if a REALLY BIG NUMBER was prime. You start plugging in inputs whose outputs are close to your REALLY BIG NUMBER until you get the result. It wouldn't even have to be a continuous function, just ordered.

So I think i am using a much more limited definition of function than you are. The sieve of eratothanes can be written in function form, but its still the sieve

Any formula that uses Sigma summation (sorry, havn't used tex in years, forgot syntax) is an algorithm, which I agree, there are lots of. And they run through countless computations to generate only a few outputs, thus making them inneffective for very large primes

I have a nice algorithm that uses the symmetry of p# and it makes the computer find primes very quickly, but it is not a function. (if your interested i'll show you, i'd like to know if its well known or perhaps something that hasn't been investigated too much anyway)

Hope this clarifies

 

[Hurkyl]

I'm not trying to be stupid, i really think this is a semantic / linguistic confusion, not an argument.

Sure -- that's why, in mathematics, we strive to define things precisely, and then do our reasoning with our precisely defined notions.

Given the usual definition of 'function', it's obvious that there exist all sorts of functions defying your limitations. For example, there is the n-th prime function. Explicitly, it's a function of the positive integers satisfying p(1) = 2 and p(n+1) = smallest prime exceeding p(n).

And there are subclasses of functions which we can prove other theorems about. For example, every (nonconstant) polynomial function with integer coefficients must take on at least one nonprime value. Similarly, if f is such a polynomial, and it satisfies f(p)=0 for every prime p, then f is the zero polynomial.

On the other hand, we have Matiyasevich's theorem which implies that there exists a polynomial g in many variables with integer coefficients for which
g(p, x_1, x_2, \cdots, x_k) = 0
has an integer solution for the x's if and only if p is prime.


If you want to say interesting things about some subclass of functions, you are going to have to find some way to express precisely what sorts of functions you want to talk about, otherwise your posts won't really have much content.

 

[CRGreathouse]

So I think i am using a much more limited definition of function than you are.

Yes, and I'm not clear on what your definition is. First of all, though, can we stop using the term "function" in that sense? I'll refer to your meaning as a thompson-function instead (and qualify it with "integer", "real", "complex", or "quaternion" as needed). Here's my first go at a definition:

The class of integer thompson-functions is the least class of functions f:\mathbb{Z}\to\mathbb{Z} including

• f(n) = 1
• f(n)=n
• f(n)=g(n)\star h(n) where g and h are integer thompson-functions and \star is exponentiation, multiplication, integer division, subtraction, or addition
• f(n)=g(h(n)) where g and h are integer thompson-functions

This allows functions like 5n^2 + 7 but not n!. How close did I come to your meaning of "function"?

Once we hash that out, you can explain the meaning of your conjecture: "All I mean is, there is no function F(M)=P for some input M that guarentees P is prime, and M is any single valued input, be it real, integer, complex or quaternian. If there where, it would be bloody famous and there'd be no need for prime testing algorithms. Especially if it was an invertable function!"

My guesses:

thompson's Conjecture (rev. 1): There exists no integer thompson-function f for which f(n) is prime for all n > 0.
thompson's (weak) Thesis (rev. 1): An integer thompson-function violating thompson's Conjecture would be strictly more efficient than any prime-listing algorithm that cannot be expressed as an integer thompson-function.
thompson's (strong) Thesis (rev. 1): A quaternion thompson-function violating a version of thompson's Conjecture would be strictly more efficient than any prime-listing algorithm that cannot be expressed as a quaternion thompson-function.

 

[thompson03]

My, this is quite the group of sarcastic and antagonistic people.

I'm so sorry that the point has been lost in favor of nitpicking semantics.

I will submit my last post with these quotes;

Prime numbers can be generated by sieving processes (such as the sieve of Eratosthenes), and lucky numbers, which are also generated by sieving, appear to share some interesting asymptotic properties with the primes. Prime numbers satisfy many strange and wonderful properties. Although there exist explicit prime formulas (i.e., formulas which either generate primes for all values or else the nth prime as a function of n), they are contrived to such an extent that they are of little practical value.

Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163).

But if you mean a polynomial or some other simple formula, then no,
there is no such formula. I am quite sure it has been proven that
there is no polynomial whose values are exactly the prime numbers. If
you want a similar proof for other kinds of simple formulas, then you
need to specify what kinds of formulas you are talking about. (For
example, you want to make sure you rule out things like p_n and
related functions.)

But there are formulas for prime numbers, and sums of prime numbers,
and sums of their logs, and such things, that have to do with
complicated transcendental functions like the Riemann Zeta Function


If you wish to argue these, please sent your argument to

http://mathworld.wolfram.com/PrimeNumber.html
Prime Number -- from Wolfram MathWorld

and

http://mathforum.org/library/drmath/view/68246.html
Math Forum - Ask Dr. Math

I for one, have had enough.

Thank you so much for enlightening me to the attitudes of this particular community.

 

[huba]

My, this is quite the group of sarcastic and antagonistic people.

I do not agree.
Greathouse has been trying to help.

As far as I am concerned, I find it very valuable that anybody interested in maths, at whatever level, has the chance in this forum to ask questions and make comments, with a good chance that they will receive help or constructive criticism.

 

[CRGreathouse]

My, this is quite the group of sarcastic and antagonistic people.

I'm not being sarcastic and I'm not antagonizing you. I'm just trying to define, precisely, what you mean by a function.

I can't speak for the true feelings of the others on this thread, but I have not sensed any antagonism from anyone here -- though Hurkyl seems to be somewhat frustrated by the lack of a clear definition.

I'm so sorry that the point has been lost in favor of nitpicking semantics.

I thought you had said this was a question of semantics?

Although there exist explicit prime formulas (i.e., formulas which either generate primes for all values or else the nth prime as a function of n), they are contrived to such an extent that they are of little practical value.

Yes, I agree with that. I did see a paper once that had an explicit formula that had been optimized for computation, but it was still quite slow compared to the best-known algorithms.

Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163).

I would argue that since the time of Euler we have penetrated much of this mystery -- though only a tiny fraction of the way toward the kind of understanding we have of, say, Euclid's geometry.

But that's getting a bit toward the philosophical, yes?

But if you mean a polynomial or some other simple formula, then no,
there is no such formula. I am quite sure it has been proven that
there is no polynomial whose values are exactly the prime numbers. If
you want a similar proof for other kinds of simple formulas, then you
need to specify what kinds of formulas you are talking about. (For
example, you want to make sure you rule out things like p_n and
related functions.)

There is no polynomial P(x) where P(1) = 2, P(2) = 3, and in general P(n) = p_n. But there are polynomials which take on positive values that are exactly the prime numbers.

 

[CRGreathouse]

I do not agree.
Greathouse has been trying to help.

Thanks for the vote of confidence. I thought I put together a reasonable post. I thought about the features thompson03 wanted in his functions, worked out a simple axiomatization, and worried that Hurkyl might be able to show by a simple example that it didn't have the properties I claimed for it. :D I'm still not entirely happy with it; nothing prohibits the function 0/0 = (1 - 1) / (1 - 1) from being defined, for example. But I wanted to get an example up more than I wanted to polish it, because perhaps thompson03 would think that some features were too much (exponentiation? integer division?) or that it left out important tools (trig functions, certain special functions to make up for the lack of series) that made the results too weak.

And I know that the conjectures I guessed at were too strong: for any reasonable definition f(n) = 2 is a function which always evaluates to a prime, so perhaps 'nonconstant function' is needed? But I wanted thompson03 to explain these restrictions himself since he, better than we, know what he's looking for.

 

[DeaconJohn]

I do not agree.
Greathouse has been trying to help.

As far as I am concerned, I find it very valuable that anybody interested in maths, at whatever level, has the chance in this forum to ask questions and make comments, with a good chance that they will receive help or constructive criticism.

Greathouse, Hubba has "hit the nail on the head," here.

You are to be commended for your patient effort to help lift Mr. Thompson to a higher level of understanding of modern efforts to find formulas for prime numbers.

Your experience with Mr. Thompson is indicative of the danger that a person submits himself to when he (or she) decides to "follow a call" to do something in this life with the hope that it will make a positive difference in the world. Even if that "call" comes in the form of a call to help others better understand and better appreciate the beauty of mathematics.

It is the "call" that makes the difference between a dull teacher and a great teacher. A teacher who is following a "call" will combine a love for his (or her) subject with a desire to communicate that love to others. Your contributions to this board clearly demonstrate that you are one of those individuals who have received such a "call," whether or not you are personally aware of it.

But, I do not mean to "put Mr. Thompson down" or to treat him with any lack of due respect. Everybody is "called" to treat every other human being with respect, and, personally, I take that as one of my highest priorities to do just that. I do not always succeed of course, but, at least I try.

On the other hand, there are a lot of people, sometimes including even me, that feel personally attacked when one of their cherished ideas (like "I am good at math and I know a lot about it") are attacked. There is no doubt that I suffered from this failing much more when I was younger, so I have a lot of sympathy for those who fall into this trap.

An unfortunate result of this situation is that the student (or the person who has put himslef in the position of student by his vast display of his own ignorance - like Mr. Thompson) rather than understanding the good intentions of the teacher, actually attacks the teacher and accusses him (or her) of being "sarcastic," "unfelling," "snotty," "pompous," "pedantic," "contolling," or whatever negatively overtonned term happens to pop into their mind. Do not think for a second that by stating this truth that I am putting myself above these people. I have made this mistake, and I continue to make this mistake, so I myself am one of these people! Naturally, it is helpful to me to be aware of this problem, and, when I do find myslef doing it, I do my best to apologize as best I can and as quickly as I can. This situation is simply one more example of the fact that nobody is perfect.

Coming under these kind of attackes is a serious danger that people like you face, but there are even more serious dangers that teachers who take their jobs seriously also face.

For example, when I was a young math teacher at LSU Baton Rouge, 40 years ago, I struggled to find a reason to give a student a "D" who really should have received an "F" in a math course for high-school math teachers. After considerable work, I found a reason and gave him a "D." Unfortunately for him, the state required that he receive a "C" in the course to be able to continue teaching math in High School. He called me up one morning at my house and threatened to burn down my house when my wife and children were in it unless I pormised him on the spot that I would change his grade to a "C." (I did have the power to change grades after the course was over.) I told him that my conscience would not allow me to do that and he hung up. Fortunately, it was a empty threat. But, such threats are not always empty and I was really shook up after that telephone call.

Notice that if I had given him the "C," he would have been teaching in a math class. He would not have been teaching mathmatics, since he had such screwed-up ideas about what mathematics is all about, but, he would have been teaching something, and, much to the detriment of his students.

One of my dedicated colleagues in a similar situation did give the student a "C," and his own son ended up in that person's math class the next semester. That colleague (Professor Heron S. Collins, author of "Finite and Infinite Dimensional Linear Spaces") told that story many time to us younger teacher for the purpose of demonstrating how wrong it is to make that mistake.

All of my colleagues at the time had similar stories to tell. That is, the colleagues of mine who took their teaching seriously. There were those who never discussed their teaching so I have no way of knowing if they were serious about their teaching or not.

And, these stories were not all about high-school math teacher bummers, in fact, far from it. There are simply many dangers that lie ahead of anybody who desires to make a positive difference in this world.

DJ

 

[DeaconJohn]

... there are polynomials which take on positive values that are exactly the prime numbers.

Greathouse, I know that what you say is true, but, I think that I saw recently that even more spectacular results have been obtained in recent times.

Do you happen to know what they are?

For example, I thought I saw that there exists a polynomial whose image is exactly the set of prime numbers when the domain of the polynomial is restricted to positive integer values of its variable (or variables). Did I get that right?

[I.e., Does there exists a polynomial p and a natural number n such that

{prime numbers} = {p(x₁, ..., x_n) | (x₁, ..., x_n) is an element of Nⁿ} where N is the set of natural numbers, N = {1, 2, 3, ...}.]

 

[CRGreathouse]

He called me up one moring at my house and threatened to burn down my house when my wife and children were in it unless I pormised him on the spot that I would change his grade to a "C." (I did have the power to change grades after the course was over.)

Words fail me. I'm sorry to hear that you had to go through that.

That seems to be a strong argument that professors should not have the power to change grades after the end of the course.

For example, I thought I saw that there exists a polynomial whose image is exactly the set of prime numbers when the domain of the polynomial is restricted to positive integer values of its variable (or variables). Did I get that right?

[I.e., Does there exists a polynomial p and a natural number n such that

{prime numbers} = {p(x₁, ..., x_n) | (x₁, ..., x_n) is an element of Nⁿ} where N is the set of natural numbers, N = {1, 2, 3, ...}.]

I think I've heard of that; certainly it sounds plausible. Actually, I might be able to construct such a polynomial from the Diophantine equation I mentioned.

So the idea is that a variable p is prime exactly when a series of formulas f_1,f_2,\ldots,f_k are identically 0 for some value of the other variables. So this is the same as finding a solution to
f_1^2+f_2^2+\cdots+f_k^2=0.

Call this quantity on the left A. If p is prime, there is some assignment with A = 0; if p is composite, A is at least 1. So let's take, say,
\frac{4A}{4A-1}
which for a satisfying assignment is 0/-1 = 0 but otherwise is in (1, 4/3]. So
1-\left\lfloor\frac{4A}{4A-1}\right\rfloor
can be 1 for prime p and is 0 otherwise. Then
(p-2)\left(1-\left\lfloor\frac{4A}{4A-1}\right\rfloor\right)
can be p-2 for prime p and is 0 otherwise. Then voila!
2+(p-2)\left(1-\left\lfloor\frac{4A}{4A-1}\right\rfloor\right)
can be p for prime p and is 2 otherwise.

Since there is at least one satisfying assignment for each prime p, this gives the polynomial you wanted.

 

[DeaconJohn]

Words fail me. I'm sorry to hear that you had to go through that.

Why, thank you Greathouse. I was pretty shook up. I actually felt better two years later when we moved to Rochester, New York. That was when I finally "put it behind me."

That seems to be a strong argument that professors should not have the power to change grades after the end of the course.

Thank you for the sympathetic response. However, it does seem to me that it is desirable for professors, instructors (I was an instructor at the time), and other teachers to have that power.

Too many times students are involved in unavoidable situations and it only makes sense to let the teachers "help out" - by letting them take the final exam the next semester - or whatever - if the teachers feel it is appropriate.

In addition, sometimes there is a real "mistake," like copying down one students grade for the grade of another student, that is not discovered right away and needs to be corrected.

And, there are so many other "dangers" or "unfortunate situations" that it seems "out of place" to plug up just one of them - and a relatively minor one at that.

I think I've heard of that; certainly it sounds plausible. Actually, I might be able to construct such a polynomial from the Diophantine equation I mentioned.

Thanks. Your construction looks good to me, though I am far from an expert in this area. Nevertheless, the techniques you use in your construction look very similar to the techniques I have read about in constructing such polynomials. That is also an indication that your construction is correct.

 

[CRGreathouse]

Thank you for the sympathetic response. However, it does seem to me that it is desirable for professors, instructors (I was an instructor at the time), and other teachers to have that power.

As a non-educator, I have no feelings about the value of letting instructors change grades late. But it seems very much like letting the bank manager open the vault: if everyone knows that he can do that, the manager and the manager's family is at risk for assault/kidnap/etc. Obviously the problem is less bad for educators, but the situations are similar.

Thanks. Your construction looks good to me, though I am far from an expert in this area. Nevertheless, the techniques you use in your construction look very similar to the techniques I have read about in constructing such polynomials. That is also an indication that your construction is correct.

Heh, just a bit of inspiration there.

 

[slider142]

For example, I thought I saw that there exists a polynomial whose image is exactly the set of prime numbers when the domain of the polynomial is restricted to positive integer values of its variable (or variables). Did I get that right?

[I.e., Does there exists a polynomial p and a natural number n such that

{prime numbers} = {p(x₁, ..., x_n) | (x₁, ..., x_n) is an element of Nⁿ} where N is the set of natural numbers, N = {1, 2, 3, ...}.]

You can use the Lagrange interpolation formula to get a polynomial of degree n that passes through n+1 specific points. Just input the first n+1 natural numbers and the first n+1 primes. :)

 

[DeaconJohn]

Slider142,

That's very true, Slider. Thank you for mentioning it. It's a little more difficult to get one that pass through all the primes. Give it a try! If you have any trouble, take a peak at the one Greathouse came up with! I think it's pretty neat.

Yours,

DJ

 

[CRGreathouse]

You can use the Lagrange interpolation formula to get a polynomial of degree n that passes through n+1 specific points. Just input the first n+1 natural numbers and the first n+1 primes. :)

Yes, but that only works for a finite collection of primes.