What makes a good teacher? What is a formula?

[Bill Spight]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

[AloneAgainstAll]

Basically to Zermelo:

You really think this is good paraphrase? I dont think so. There are things we cant "enclose" in mathemathical formulas (like results of integral of sin(x^2) cant be described by elementary functions, only infinite sum ) etc. So the fact that we can produce such a formula (even if its completely useless for computing) is mathemathically significant.

Ok, I define S(t) as the integral of sin(x^2) from 0 to t. Now I can write you a nice short correct formula for it, i.e. S(t). Yes, it does not use elementary functions only, but your prime formula with floor functions does not use only elementary functions either. The whole issue is really about what kind of formulas we allow.

I don't dispute that maybe you had a bad teachers, and maybe he/she could have discussed the nuances of formulas and algorithms better.

Its not "mine" formula, its Paulo Ribenboim formula i believe (or maybe someone other, not sure). If you look closely at your example you will find a small mistake there, but i know what you mean (thats completely sidenote, you dont need to improve your example, i trust you know the drill)

Yes this formula use floor function which is not elementary function. But that is completely another topic, i used this example with S(x^2) to explain (i guess i was not understood) that enclosing sth in formula is mathematically significant, even if formula is not usefull for computations.

You ask what formulas we allow? All that are well defined in ZFC (+AC or not, as you wish). If we would try to limit it, then why to make any formulas at all?

You really think this is about what formulas we allow? I dont think so. Now we are wise, we know that such a formulas exist (though they are not usefull for computations, not important), but can you provide formula which would produce nontrivial zeros for Riemmann-Zeta function? I guess no. But imagine that in 10 years mathematicians broke this, computed all nontrivial zeros and proved that there are no more. Now producing such a formula looks pretty different, isnt it?

She couldnt explain nuances of formulas and algorithms. She was elementary school teacher, and i was 10 years old then.

[AloneAgainstAll]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

I think your attitude toward me is in contradiction with forum rules - you bash me all the time, completely disrespect me, play dumb and catch on grammar mistakes and niuances. Do you feel good with that?

[Bill Spight]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

I think your attitude toward me is in contradiction with forum rules - you bash me all the time, completely disrespect me, play dumb and catch on grammar mistakes and niuances. Do you feel good with that?

IOW, you don't know.

[AloneAgainstAll]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

I think your attitude toward me is in contradiction with forum rules - you bash me all the time, completely disrespect me, play dumb and catch on grammar mistakes and niuances. Do you feel good with that?

IOW, you don't know.

You think you are being funny?

[Bill Spight]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

I think your attitude toward me is in contradiction with forum rules - you bash me all the time, completely disrespect me, play dumb and catch on grammar mistakes and niuances. Do you feel good with that?

IOW, you don't know.

You think you are being funny?

I think I am being accurate.

[AloneAgainstAll]

So are there anymore backup for your claims, or thats all?

[kj01a]

This is my favorite math forum <3

[Bill Spight]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

So are there anymore backup for your claims, or thats all?

You posted an expression which you claim is one of several formulae that work. Apparently you are unaware of a simpler one. OK.

Please explain what the expression means. You may assume that we are elementary school students.

[Kirby]

Wilson's theorem here is simpler, in my opinion: https://en.wikipedia.org/wiki/Formula_for_primes

But it's not that useful. You can express many computer algorithms in the form of a "formula", but it's not necessarily that useful - it's just a way to express the relationship.

Note that in the article linked above, there's a proof showing that no non-constant polynomial P(n) with integer coefficients exists that evaluates to a prime number for all integers n. Maybe something along these lines is what the professor was getting at.

Anyway, back to the point, I agree that there can be bad teachers. As a student, it's probably in your best interest to strive for success despite this.

[Kirby]

This discussion has shown me how rusty my math is. It'd be good to get back into studying math, but it's a little hard to prioritize it. For job stuff, it's more practical to directly study stuff related to my job, for example.

I kind of want to get back into studying math, but it'd be purely for fun. And if I'm going to do that, I guess it's more practical to study go or something...

[AloneAgainstAll]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

So are there anymore backup for your claims, or thats all?

You posted an expression which you claim is one of several formulae that work. Apparently you are unaware of a simpler one. OK.

Please explain what the expression means. You may assume that we are elementary school students.

You always answer a question with a question? You need to make some investigation yourself - great Honinbo Shuei meijin advice always on props.
You claimed that such a formula does not exist - its high time to back your claim with some proof, or at least prove that formula i showed is wrong.

@Kirby

If you look closely at wiki page, then you will see difference between "formula that generate prime numbers" with "formula that for all natural n, f(n)=n-th prime number". But neverthelss Wilson achievement is impressive and elegant, no doubt about that.

[Bill Spight]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

IOW, you don't know.

So are there anymore backup for your claims, or thats all?

You posted an expression which you claim is one of several formulae that work. Apparently you are unaware of a simpler one. OK.

Please explain what the expression means. You may assume that we are elementary school students.

You always answer a question with a question? You need to make some investigation yourself - great Honinbo Shuei meijin advice always on props.
You claimed that such a formula does not exist - its high time to back your claim with some proof, or at least prove that formula i showed is wrong.

Your elementary school teacher made what is, at worst, an ambiguous claim, which depends upon what is a formula. It was a claim that also had been made by mathematicians. You then said

I wouldnt reccomend that math teacher to even my enemies.

I defended your teacher, who I still believe was right, based upon her understanding of what a formula is. My guess is that it was something like this. Given the Nth prime, Pn, and other information that we already know, which in this case would presumably include the identities of the first N-1 primes, but would not include information about higher numbers, such a formula is an expression which, when evaluated, will produce the N+1th prime. (If we are not restricted to what we already know, then we can find the next prime by examining numbers greater than Pn. The point of having a formula is to leap from the known to the unknown.)

The expression you provide requires inspecting numbers up to 2^(N+1) to find the N+1th prime. It would not be such a formula, based upon that criterion.

Your turn. Explain your expression, please.

[Bill Spight]

Rigor you can maintain?

mortis.

[Kirby]

Continuing the off-topic math discussion:

This discussion has shown me how rusty my math is. It'd be good to get back into studying math, but it's a little hard to prioritize it. For job stuff, it's more practical to directly study stuff related to my job, for example.

I kind of want to get back into studying math, but it'd be purely for fun. And if I'm going to do that, I guess it's more practical to study go or something...

Funnily enough, with this talk about math, I went ahead and finished a book that I'd been inching my way through, bit by bit. It's called "How Not to Be Wrong" (https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535). All in all, I'd recommend the book. The only reason I've been going through it slowly is that I divide my reading time up between different books (I'm working on about 3 other books in parallel - a little weird, I know).

Anyway, I enjoyed the concluding statement Ellenberg writes in the book, which I'll leave in hide tags in case anyone else is going to read it:

Under the last section of the book, subtitled, "When Am I Going To Use This?":

Every time you observe that more of a good thing is not always better; or you remember that improbable things happen a lot, given enough chances, and resist the lure of the Baltimore stockbroker; or you make a decision based not just on the most likely future, but on the cloud of all possible futures, with attention to which ones are likely and which ones are not; or you let go of the idea that the beliefs of groups should be subject to the same rules as beliefs of individuals; or, simply, you find that cognitive sweet spot where you can let your intuition run wild on the network of tracks formal reasoning makes for it; without writing down an equation or drawing a graph, you are doing mathematics, the extension of common sense by other means. When are you going to use it? You've been using mathematics since you were born and you'll probably never stop. Use it well.

So perhaps my statement above is a little shortsighted: I'm already doing math in my every day life; perhaps just not as well as I could be.

Does that make formal study worthy of my limited time? I'm not exactly sure, yet, but I lean more toward saying yes than before...