20200211, 03:41  #1 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5×7×83 Posts 
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? 
20200211, 16:31  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{3}×1,223 Posts 
Is this a puzzle or just an attempt to get someone else to do work for you?

20200211, 19:58  #3 
"Curtis"
Feb 2005
Riverside, CA
2^{8}×19 Posts 
If it takes more than a CPUday, Sweety calls it a conjecture and hopes someone else does the work.

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Smallest k>1 such that Phi_n(k) is prime  sweety439  sweety439  12  20210628 09:24 
Smallest prime of the form a^2^m + b^2^m, m>=14  JeppeSN  Math  114  20181216 01:57 
Smallest prime with a digit sum of 911  Stargate38  Puzzles  6  20140929 14:18 
Smallest floor of k for cullen prime  Citrix  Prime Cullen Prime  12  20070426 19:52 
Smallest tenmilliondigit prime  Heck  Factoring  9  20041028 11:34 