It seems that small [URL="https://stdkmd.net/nrr/repunit/phin10.htm"]factors of Phi_n(10)[/URL] are searched for all n around 100000 and 200000, but small factors of Phi_n(2) are only searched by prime n and the n's which are power of 2, I know this project is searching this for n's which are twice an odd prime, bur how about other n? Is there anyone searching small factors of Phi_n(2) for all n around 100000 and 200000?

The "[URL="https://oeis.org/A250197/a250197_2.txt"]factors of Phi_n(2) for n<=1280[/URL]" file is subject to updates as new factors are found.

Because of the way OEIS handles files, every time the contents are updated, the link changes. The latest link is now _2.txt, which differs from _1.txt by having additional factors for exponents 991, 1213, 1219, 1261.

[QUOTE=GP2;519951]The "[URL="https://oeis.org/A250197/a250197_2.txt"]factors of Phi_n(2) for n<=1280[/URL]" file is subject to updates as new factors are found.

Because of the way OEIS handles files, every time the contents are updated, the link changes. The latest link is now _2.txt, which differs from _1.txt by having additional factors for exponents 991, 1213, 1219, 1261.[/QUOTE]

Well, can you extend this list to n=2000 or above?

[QUOTE=sweety439;520008]Well, can you extend this list to n=2000 or above?[/QUOTE]

The recent factors for those four exponents were readily available from the Cunningham tables.

If you want exponents beyond the Cunningham range, you could probably gather the data from FactorDB.com

[QUOTE=GP2;520018]The recent factors for those four exponents were readily available from the Cunningham tables.

If you want exponents beyond the Cunningham range, you could probably gather the data from FactorDB.com[/QUOTE]

However, you cannot enter "Phi_n(2)" in factordb, since factordb has no "cyclotomic polynomial" function, you can only enter "2^n-1" (I know that 2^n-1 = prod{d|n}Phi_d(2))

[URL="https://www.mersenne.ca/exponent/1073741827"]M1,073,741,827[/URL] has a factor: 16084529043983099051873383

This exponent is relevant to the (trivial) "[URL="http://mprime.s3-website.us-west-1.amazonaws.com/new_mersenne_conjecture.html"]New Mersenne Conjecture[/URL]"

[QUOTE=GP2;520259][URL="https://www.mersenne.ca/exponent/1073741827"]M1,073,741,827[/URL] has a factor: 16084529043983099051873383

This exponent is relevant to the (trivial) "[URL="http://mprime.s3-website.us-west-1.amazonaws.com/new_mersenne_conjecture.html"]New Mersenne Conjecture[/URL]"[/QUOTE]

M1,073,741,827 is Phi(2^30+3, 2)

Conjectures:

* Phi(2^n-1,2) is composite for all n>7 (it is prime for n = 2, 3, 4, 5, 7)
* Phi(2^n+1,2) is composite for all n>7 (it is prime for all n <= 7)
* Phi(2^n-3,2) is composite for all n>6 (it is prime for n = 3, 4, 6)
* Phi(2^n+3,2) is composite for all n>4 (it is prime for n = 1, 2, 4)

* Phi(2*(2^n-1),2) is composite for all n>7 (it is prime for all n <= 7)
* Phi(2*(2^n+1),2) is composite for all n>4 (it is prime for n = 1, 2, 4)
* Phi(2*(2^n-3),2) is composite for all n>6 (it is prime for n = 2, 3, 4, 6)
* Phi(2*(2^n+3),2) is composite for all n>4 (it is prime for n = 1, 2, 3, 4)

* There are no odd n>345 such that both Phi(n,2) and Phi(2*n,2) are primes (there are both primes for n = 3, 5, 7, 13, 15, 17, 19, 31, 49, 61, 85, 127, 345, only consider odd n)
* There are no odd n>345 such that both Phi(n,2)/gcd(Phi(n,2),n) and Phi(2*n,2)/gcd(Phi(2*n,2),n) are primes (there are both primes for n = 5, 7, 9, 13, 15, 17, 19, 21, 27, 31, 49, 61, 85, 127, 345, only consider odd n)

Related to the New Mersenne Conjecture

The conjecture that there are only 5 Fermat primes is that there are no n>5 such that Phi(2^n,2) is prime, I conjectured that there are no n>7 such that Phi(2^n-1,2) is prime, no n>7 such that Phi(2^n+1,2) is prime, no n>7 such that Phi(2*(2^n-1),2) is prime, and no n>4 such that Phi(2*(2^n+1),2) is prime.

More generally, for every (positive or negative or zero, odd or even) integer k, there are only finitely many n such that 2^n+k is in [URL="https://oeis.org/A072226"]OEIS A072226[/URL], i.e. there are only finitely many n such that Phi(2^n+k,2) is prime.

[CODE]
k conjectured full list of such n
-16 5,
-15 5, 6,
-14 4,
-13 4, 5,
-12 4,
-11 4, 11,
-10 4, 5,
-9 4,
-8 4, 5, 6, 7,
-7 4,
-6 3, 4, 5, 7,
-5 3, 5, 9,
-4 3, 4,
-3 3, 4, 6,
-2 2, 3, 4, 5, 6, 7, 8,
-1 2, 3, 4, 5, 7,
0 1, 2, 3, 4, 5,
1 1, 2, 3, 4, 5, 6, 7,
2 1, 2, 3, 5,
3 1, 2, 4,
4 1, 2, 3,
5 1, 2, 3, 6, 7, 8,
6 1, 2, 3, 4, 5,
7 1, 3,
8 1, 2, 3, 4, 5,
9 2, 3, 9,
10 1, 2, 4, 5,
11 1, 2, 3, 4,
12 1, 2,
13 1, 2, 6,
14 1, 3, 4, 5, 6,
15 1, 2, 4,
16 3, 4, 6,
[/CODE]

[CODE]ECM found a factor in curve #8, stage #2
Sigma=5999343673417650, B1=50000, B2=5000000.
2^114743+1 has a factor: 363690536981293584763 (ECM curve 8, B1=50000, B2=5000000)[/CODE]
W114743 = 565224019 · 581747011 · 601253321 · 810315067 · 69667542321371 · 7485151305966881 · 13863811976194993 · 363690536981293584763 · PRP34439

[QUOTE=axn;520425][CODE]ECM found a factor in curve #8, stage #2
Sigma=5999343673417650, B1=50000, B2=5000000.
2^114743+1 has a factor: 363690536981293584763 (ECM curve 8, B1=50000, B2=5000000)[/CODE]
W114743 = 565224019 · 581747011 · 601253321 · 810315067 · 69667542321371 · 7485151305966881 · 13863811976194993 · 363690536981293584763 · PRP34439[/QUOTE]

Congrats :toot:

[CODE]time echo 'print((2^114743+1)/3/565224019/581747011/601253321/810315067/69667542321371/7485151305966881/13863811976194993/363690536981293584763)' | gp -q | ./bpsw-2 - 1 2 114743 1
Testing (2*x)^((n + 1)/2) == -2 (mod n, x^2 - 9*x + 1)...
Likely prime!

real 0m8.926s
user 0m8.980s
sys 0m0.000s

[/CODE]

Well within reach of a Primo proof. Edit: A hard proof for Primo -- I was thinking base 2!

[QUOTE=paulunderwood;520432]Well within reach of a Primo proof.[/QUOTE]
This would go as #2 on [url]http://www.ellipsa.eu/public/primo/top20.html#PrimoRecord[/url] (if it were to be attempted).

How many core years do you estimate to prove this one?

EDIT:- Just missed paul's edit. The question still stands.