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 2019-06-23, 14:05 #298 sweety439     Nov 2016 40558 Posts It seems that small factors of Phi_n(10) 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?
 2019-06-24, 04:52 #299 GP2     Sep 2003 2,579 Posts The "factors of Phi_n(2) for n<=1280" 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.
2019-06-24, 21:07   #300
sweety439

Nov 2016

7×13×23 Posts

Quote:
 Originally Posted by GP2 The "factors of Phi_n(2) for n<=1280" 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.
Well, can you extend this list to n=2000 or above?

2019-06-25, 03:21   #301
GP2

Sep 2003

2,579 Posts

Quote:
 Originally Posted by sweety439 Well, can you extend this list to n=2000 or above?
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

2019-06-25, 16:56   #302
sweety439

Nov 2016

7×13×23 Posts

Quote:
 Originally Posted by GP2 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
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))

 2019-06-28, 15:00 #303 GP2     Sep 2003 50238 Posts M1,073,741,827 has a factor: 16084529043983099051873383 This exponent is relevant to the (trivial) "New Mersenne Conjecture"
2019-06-29, 00:02   #304
sweety439

Nov 2016

40558 Posts

Quote:
 Originally Posted by GP2 M1,073,741,827 has a factor: 16084529043983099051873383 This exponent is relevant to the (trivial) "New Mersenne Conjecture"
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

Last fiddled with by sweety439 on 2019-06-29 at 00:22

 2019-06-29, 00:11 #305 sweety439     Nov 2016 7×13×23 Posts 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 OEIS A072226, 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, Last fiddled with by sweety439 on 2019-06-29 at 00:21
 2019-07-01, 03:55 #306 axn     Jun 2003 4,637 Posts 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) W114743 = 565224019 · 581747011 · 601253321 · 810315067 · 69667542321371 · 7485151305966881 · 13863811976194993 · 363690536981293584763 · PRP34439
2019-07-01, 05:35   #307
paulunderwood

Sep 2002
Database er0rr

11×13×23 Posts

Quote:
 Originally Posted by axn 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) W114743 = 565224019 · 581747011 · 601253321 · 810315067 · 69667542321371 · 7485151305966881 · 13863811976194993 · 363690536981293584763 · PRP34439
Congrats

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

Last fiddled with by paulunderwood on 2019-07-01 at 05:44

2019-07-01, 05:45   #308
axn

Jun 2003

4,637 Posts

Quote:
 Originally Posted by paulunderwood Well within reach of a Primo proof.
This would go as #2 on http://www.ellipsa.eu/public/primo/t...ml#PrimoRecord (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.

Last fiddled with by axn on 2019-07-01 at 05:46

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