mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2020-02-11, 03:41   #1
sweety439
 
sweety439's Avatar
 
Nov 2016

2×11×103 Posts
Default Smallest k>1 such that Phi_n(k) is prime

I found the smallest k>=2 such that Phi_n(k) is (probable) prime (where Phi is the cyclotomic polynomial) for all 1<=n<=2500, see the text file (file format: "n k"). The k has been searched for special value of n's, see these OEIS sequences.

A066180 (for prime n)
A103795 (for n=2*p with p odd prime)
A056993 (for n=2^k with k>=1)
A153438 (for n=3^k with k>=2)
A246120 (for n=2*3^k with k>=1)
A246119 (for n=3*2^k with k>=1)
A298206 (for n=9*2^k with k>=1)
A246121 (for n=6^k with k>=1)
A206418 (for n=5^k with k>=2)
A205506 (for n=6*2^i*3^j with i,j>=0)
A181980 (for n=10*2^i*5^j with i,j>=0)

Let a(n) be the smallest k>=2 such that Phi_n(k) is prime, I found a(n) for all 1<=n<=2500, and according to these sequences, a(2^n) is known for all 0<=n<=21, a(3^n) is known for all 0<=n<=11, a(2*3^n) is known for all 0<=n<=10, etc. and the k's for some large n are a(2^21)=919444, a(3^12)=94259, a(2*3^11)=9087, etc. However, it seems that there is no project for finding a(n) for general n. (this a(n) is the OEIS sequence A085398)

Conjecture, for all n>=1, there exists k>=2 such that Phi_n(k) is prime. (if this conjecture is true, then there are infinitely many such k for all n>=1, besides, this conjecture is true if Bunyakovsky conjecture is true)

Can someone find the smallest k>=2 such that Phi_n(k) is (probable) prime (where Phi is the cyclotomic polynomial) for 2501<=n<=10000? Or larger n?
Attached Files
File Type: txt least k such that phi(n,k) is prime.txt (22.0 KB, 52 views)
sweety439 is offline   Reply With Quote
Old 2020-02-11, 16:31   #2
Uncwilly
6809 > 6502
 
Uncwilly's Avatar
 
"""""""""""""""""""
Aug 2003
101×103 Posts

26·33·5 Posts
Default

Is this a puzzle or just an attempt to get someone else to do work for you?
Uncwilly is online now   Reply With Quote
Old 2020-02-11, 19:58   #3
VBCurtis
 
VBCurtis's Avatar
 
"Curtis"
Feb 2005
Riverside, CA

103458 Posts
Default

If it takes more than a CPU-day, Sweety calls it a conjecture and hopes someone else does the work.
VBCurtis is online now   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Smallest k>1 such that Phi_n(k) is prime sweety439 And now for something completely different 9 2020-06-03 18:11
Smallest prime of the form a^2^m + b^2^m, m>=14 JeppeSN Math 114 2018-12-16 01:57
Smallest prime with a digit sum of 911 Stargate38 Puzzles 6 2014-09-29 14:18
Smallest floor of k for cullen prime Citrix Prime Cullen Prime 12 2007-04-26 19:52
Smallest ten-million-digit prime Heck Factoring 9 2004-10-28 11:34

All times are UTC. The time now is 16:10.

Wed Sep 30 16:10:41 UTC 2020 up 20 days, 13:21, 0 users, load averages: 1.97, 2.11, 1.95

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.