Riesel/Sierp #'s for bases 3, 7, and 15

[robert44444uk]

Originally Posted by R. Gerbicz

I've found by my program this new record Sierpinski value for base=3:
1694420352676*3^n+1 is composite for all n values. The covering set is
5,7,13,19,37,41,73,757,6481

new record Sierpinski value for base=7:
1112646039348*7^n+1 is composite for all n values. The covering set is
5,13,19,37,43,73,181,193,1201

Totally great! Such a breathtaking reduction in base 3.

[michaf]

Originally Posted by michaf

Congrats and thanks :)
base 7 is a magnitude harder still, due to the larger base, but also, doable within a few thousand cpu-years (I'll give it a go to test how much will be left for base 7)

Hmm.. I can't seem to get the modular restricions in as easy as I thought...

div=powmod(k,1,3);
if(div=1) {
k+=2;
goto start;
}

doesn't work :(

[R. Gerbicz]

Again, new record for Sierpinski, base=3:
1125458784774*3^n+1 is composite for all n values.
Covering set=5,7,13,17,19,37,41,73,193,757

[KEP]

Originally Posted by R. Gerbicz

Again, new record for Sierpinski, base=3:
1125458784774*3^n+1 is composite for all n values.
Covering set=5,7,13,17,19,37,41,73,193,757

Wow another reduction by half a trillion k's , really awesome what you accomplish here, this is really helpfull, wonder if you actually overtakes my effort on riesel base 3, if you magically comes up with another reduction ...

Now I'm just wondering, is there someway to make this a distributed effort, in order to find more covering sets or can the tast of finding covering sets only be at one machine at a time? Also how do you start this search, from lowest k and up or from conjectured k and then goes down?

Regards

Kenneth!

[R. Gerbicz]

Originally Posted by KEP

Now I'm just wondering, is there someway to make this a distributed effort, in order to find more covering sets or can the tast of finding covering sets only be at one machine at a time? Also how do you start this search, from lowest k and up or from conjectured k and then goes down?

New record for Sierpinski base=3 !
125050976086*3^n+1 is composite for all n.
covering set=5,7,13,17,19,37,41,193,757

Small correction:
For the above posted Sierpinski base=7 record k=1112646039348
this is a smaller covering set: 5,13,19,43,73,181,193,1201

Now you can also enjoy the search! (I've stopped it.)

I've uploaded my code, you can download it: http://robert.gerbicz.googlepages.com/covering.c
And an exe optimized by flags for P4: http://robert.gerbicz.googlepages.com/covering.exe

The program requires 5 integers to start:
exponent base C primebound best

where we are testing exponent, it means that the period of the covering set's length will be this number (or it's divisor),
known good examples are those where this number has got lots of small divisors, say exponent=24,72,144,...

base is the base of the sequence

C is 1 for Sierpinski, -1 for Riesel, it means we are testing k*b^n+C sequence (it isn't interesting, but
you can use other values also)

primebound: up to this number we consider all primes which divides b^exponent-1, I've used 10000,
you can use larger/smaller values for it, but very large, say 1000000 is obviously inefficient and slow down the program

best: upper bound for the k value, we are searching k values for that k<best.
It's good to set it to the best knwon k+1.

Note that the product of the last two parameters should be < 2^62, otherwise it'll be an integer overflow.
(For our search it isn't very interesting.)
I think up to base<2^15 the program is good.


Here are some examples to (re)discover currently known record solutions:
24 15 1 10000 100000000000000
find in 1 second k=91218919470156 for exponent=24,
base=15, 1 so Sierpinski, prime bound=10000, best k=100000000000000

24 7 1 10000 2000000000000
find in 1 second k=1112646039348 for exponent=24,
base=7, 1 so Sierpinski, prime bound=10000, best k=2000000000000

144 3 1 10000 126000000000
find (this took about half an hour or so) k=125050976086 for exponent=144,
base=3, 1 so Sierpinski, prime bound=10000, best k=126000000000

24 7 -1 10000 410000000000
find in 1 second k=408034255082 for exponent=24,
base=7, -1 so Riesel, prime bound=10000, best k=410000000000

[robert44444uk]

Robert

I am so happy you have posted a windows executable! I am planning to research base 3 some more.

Do you have any timings for your programme when you get into larger "exponents" such as 330 or 2310, as 3^11-1 brings in two smallish primes, 23 and 3851?

[R. Gerbicz]

Originally Posted by robert44444uk

Robert

I am so happy you have posted a windows executable! I am planning to research base 3 some more.

Do you have any timings for your programme when you get into larger "exponents" such as 330 or 2310, as 3^11-1 brings in two smallish primes, 23 and 3851?

It's hard to predict the timing. I would try only those exponents, which are divisible by 24, all recently found record solutions have period length divisible by 24! So 330,2310 aren't a very good run. Yes, they bring in 23, but lots of small primes are excluded from the covering set, see the listed primes if you run the program.

[henryzz]

Originally Posted by R. Gerbicz

It's hard to predict the timing. I would try only those exponents, which are divisible by 24, all recently found record solutions have period length divisible by 24! So 330,2310 aren't a very good run. Yes, they bring in 23, but lots of small primes are excluded from the covering set, see the listed primes if you run the program.

is there any sort of counter u can put on

[robert44444uk]

Originally Posted by R. Gerbicz

It's hard to predict the timing. I would try only those exponents, which are divisible by 24, all recently found record solutions have period length divisible by 24! So 330,2310 aren't a very good run. Yes, they bring in 23, but lots of small primes are excluded from the covering set, see the listed primes if you run the program.

But I won't feel comfortable unless all k up to n=330 or 2310 (or any other value for that matter) are tested and all remaining k are shown not to have a simple covering set.

I am nervous because CRM does provide unpredictable results, and it is possible that, with the mods all lining up, a quite low value might pop out as a solution, despite the superficially unattractive modulo order of the candidate cover set primes and mod requirements for each prime.

[R. Gerbicz]

Originally Posted by robert44444uk

But I won't feel comfortable unless all k up to n=330 or 2310 (or any other value for that matter) are tested and all remaining k are shown not to have a simple covering set.

I am nervous because CRM does provide unpredictable results, and it is possible that, with the mods all lining up, a quite low value might pop out as a solution, despite the superficially unattractive modulo order of the candidate cover set primes and mod requirements for each prime.

As I read on this topic Simelink tried all exponents up to 144 for Riesel type, for base=3,7,15 by his program. You can do it on the Sierpinski type, that wouldn't take so much time, I haven't done that, in fact I've checked only about 10 different exponents for these hard bases on the Sierpinski side. So there can be low exponents which give better k values.

[tnerual]

i'm testing sierp base 3 ... first results:

covering
144
3
1
100000
1694420352677
Checking k*3^n+1 sequence for exponent=144, bound for primes in the covering set=100000, bound for k is 1694420352677
Examining primes in the covering set: 13,5,7,41,757,73,17,193,19,37,6481,97,577,769
And their orders: 3,4,6,8,9,12,16,16,18,18,24,48,48,48
**************** Solution found ****************
1125458784774
**************** Solution found ****************
405697492212
**************** Solution found ****************
135232497404
**************** Solution found ****************
125050976086

i continue up to exponent = ? and i will edit the above quote ...