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Using calculus to do number theory
11–20 of 23 posts
Re: Using calculus to do number theory
#12In some sense the title is a bit misleading (one is led to think of Analytic Number Theory). I'd rather use the title "Using differentials to...", which is more precise, as there is not exactly any "calculus" going on but there is indeed differential algebra and differential "number theory", so to speak. But great and elegant article. Thanks.
The technical term is "formal derivative" b/c there are no limits involved, it's basically a rewrite rule for changing x^n to nx^(n-1).
Re: Using calculus to do number theory
#13This idea leads to the p-adic numbers:
Re: Using calculus to do number theory
#14In some sense the title is a bit misleading (one is led to think of Analytic Number Theory). I'd rather use the title "Using differentials to...", which is more precise, as there is not exactly any "calculus" going on but there is indeed differential algebra and differential "number theory", so to speak. But great and elegant article. Thanks.
The technical term is "formal derivative" b/c there are no limits involved, it's basically a rewrite rule for changing x^n to nx^(n-1).
I know, yes, but did not want to digress. Thanks though. Actually, it is the "extension" to K[e] with e^2=0, as "usual".
Re: Using calculus to do number theory
#15Blog looks promising. Is there a RSS feed associated with it?
Re: Using calculus to do number theory
#16In some sense the title is a bit misleading (one is led to think of Analytic Number Theory). I'd rather use the title "Using differentials to...", which is more precise, as there is not exactly any "calculus" going on but there is indeed differential algebra and differential "number theory", so to speak. But great and elegant article. Thanks.
Actually it is Calculus in p-adic numbers!
Re: Using calculus to do number theory
#17In some sense the title is a bit misleading (one is led to think of Analytic Number Theory). I'd rather use the title "Using differentials to...", which is more precise, as there is not exactly any "calculus" going on but there is indeed differential algebra and differential "number theory", so to speak. But great and elegant article. Thanks.
Actually it is Calculus in p-adic numbers!
Mmmmhhhh, sure? Because p-adic numbers have characteristic 0, AFAIK.
Re: Using calculus to do number theory
#18Earlier quoted context omitted.
The technical term is "formal derivative" b/c there are no limits involved, it's basically a rewrite rule for changing x^n to nx^(n-1).
I know, yes, but did not want to digress. Thanks though. Actually, it is the "extension" to K[e] with e^2=0, as "usual".
The infinitesimally thickened point.
Re: Using calculus to do number theory
#19I love complex analysis, and that's the branch of calculus that is most associated with number theory. For example, it was critical in the original proof of the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. Today, all kinds of number theoretic functions are studied using complex analysis, like the famous Riemann zeta function, Dirichlet L-functions, theta functions, and so on. So…
>for any y >= n with f(y) = 0 (mod n), there's some x between 0 and n-1
There's a simpler way to see this, any such y can be represented as y = nk + x where i,j are divisor & remainder. Then f(y) = f(nk + x) = f(x) modulo n since by binomial theorem all other terms other than those with just x will be divisible by n.
Re: Using calculus to do number theory
#20Loved this. Thanks.