 mersenneforum.org Density of norms (field theory)
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 Register FAQ Search Today's Posts Mark Forums Read 2018-12-28, 00:09 #1 carpetpool   "Sam" Nov 2016 5178 Posts Density of norms (field theory) In this post here, I asked for the conditional probability for an integer N being prime given that all prime q dividing n are congruent to 1 modulo 2*p (for some prime p). As a result, I also got the answer of how many integers N not exceeding x can be written as a product of primes only congruent to 1 modulo 2*p. This is asymptotically D(x) = c*x*(log(x))^(1/(p-1) - 1) for some constant c, which seems to be decreasing significantly as p increases. How many integers N not exceeding x can (I) be written as a product of primes only congruent to 1 modulo 2*p and (II) in addition to (I), N can be expressed as the norm for some integral element f in the ring of integers in K=Q(zeta(p)) where K is the field of p-th roots of unity (the p-th cyclotomic field) ? The condition for (II) can be restated as there is at least one ideal of norm N that is principal in K. I am hoping for a precise answer (as in my last thread) in an attempt to solve another problem related to this. Again, any information is helpful, and thanks for help.   2018-12-28, 00:19   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

24×3×52×7 Posts Quote:
 Originally Posted by carpetpool In this post here, I asked for the conditional probability for an integer N being prime given that all prime q dividing n are congruent to 1 modulo 2*p (for some prime p). As a result, I also got the answer of how many integers N not exceeding x can be written as a product of primes only congruent to 1 modulo 2*p. This is asymptotically D(x) = c*x*(log(x))^(1/(p-1) - 1) for some constant c, which seems to be decreasing significantly as p increases. How many integers N not exceeding x can (I) be written as a product of primes only congruent to 1 modulo 2*p and (II) in addition to (I), N can be expressed as the norm for some integral element f in the ring of integers in K=Q(zeta(p)) where K is the field of p-th roots of unity (the p-th cyclotomic field) ? The condition for (II) can be restated as there is at least one ideal of norm N that is principal in K. I am hoping for a precise answer (as in my last thread) in an attempt to solve another problem related to this. Again, any information is helpful, and thanks for help.
(2kp+1)(2jp+1)=4jkp^2+2(k+j)p+1 = 2(2jkp+k+j)p+1 so as many as the natural numbers up to X/(2p) of form 2jkp+k+j for some natural numbers k and j.

Last fiddled with by science_man_88 on 2018-12-28 at 00:20   2018-12-28, 19:05 #3 carpetpool   "Sam" Nov 2016 5·67 Posts It seems to be that the answer is 0, although I don't know for sure because many of my previous posts seem to be getting lack of attention due to the little known information and research of these topics. However, I did find this article. Still, any other explanations of it are welcome.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post Cruelty Proth Prime Search 158 2020-07-31 22:23 carpetpool carpetpool 24 2017-10-29 23:47 devarajkandadai Number Theory Discussion Group 11 2017-10-28 20:58 Nick Math 4 2017-04-01 16:26 xilman Science & Technology 85 2010-12-20 21:42

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