 mersenneforum.org Smallest k>1 such that Phi_n(k) is prime
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sweety439

Nov 2016

2,267 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?
Attached Files least k such that phi(n,k) is prime.txt (22.0 KB, 52 views)   2020-02-11, 16:31 #2 Uncwilly 6809 > 6502   """"""""""""""""""" Aug 2003 101×103 Posts 26×33×5 Posts Is this a puzzle or just an attempt to get someone else to do work for you?   2020-02-11, 19:58 #3 VBCurtis   "Curtis" Feb 2005 Riverside, CA 10000111001012 Posts If it takes more than a CPU-day, Sweety calls it a conjecture and hopes someone else does the work.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 And now for something completely different 9 2020-06-03 18:11 JeppeSN Math 114 2018-12-16 01:57 Stargate38 Puzzles 6 2014-09-29 14:18 Citrix Prime Cullen Prime 12 2007-04-26 19:52 Heck Factoring 9 2004-10-28 11:34

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