| Prime | Minimum Generator | Approx log2 |
|---|---|---|
| 59656*2^59656+1 | 3 | 59671.9 |
| 32292*2^32292+1 | 5 | 32307.0 |
| 18496*2^18496+1 | 3 | 18510.2 |
| 1477!+1 | 1481 | 13426.3 |
| 1706595*2^11235+1 | 13 | 11255.7 |
| 2897*2^9715+1 | 3 | 9726.5 |
| 872!+1 | 983 | 7266.0 |
| 6611*2^6611+1 | 3 | 6623.7 |
| 4713*2^4713+1 | 5 | 4725.2 |
| 427!+1 | 467 | 3120.8 |
| 399!+1 | 409 | 2877.4 |
| 340!+1 | 347 | 2374.2 |
| 2^2281-1 | 3 | 2281 |
| 320!+1 | 367 | 2206.8 |
| 2^1279-1 | 5 | 1279 |
| 154!+1 | 163 | 901.9 |
| 2^607-1 | 5 | 607 |
Many of the entries are Cullen primes. The 211235 entry was mentioned earlier, also see the post about Mersenne Primes. It appears that the minimum primitive root of X factorial plus 1 is always greater than X.
The C18496 entry took 2.72 days on an Athlon 700, with
Pari/GP 2.1.3, most of the time which was spent on proving
its primality, even though I already knew it was prime.
Later I learned about that one can essentially assert primality of P for znprimroot with addprimes(P).
The 29715 entry came from the Sierpinski problem.
The factorial primes 427 and greater used the following gp code, which relies on the conjecture that the least primitive root is greater than n, and is prime.
m=n!; forprime(root=n, n+10000, print("root=",root); fl=0; forprime(p=2,n, if((Mod(root,m+1)^(m/p)==Mod(1,m+1)),
print(" unhappy at ", p); fl=1; break)); if(fl==0,break))
No comments :
Post a Comment