Problems & Puzzles: Problems

Problem 30.- A stepladder to Ba sequence

As we stated at the Problem 29, Ba is “the lowest Brier number”, that is to say, Ba is an odd number k such that k*2^n+1 and k*2^n-1 are composite for all n=>1. In other words Ba is the lowest number such that Ba is at the same time a Sierpinski and a Riesel number.

As probably you know, it is conjectured that the smallest Sierpinski number is 78557 and that the smallest Riesel number is 509203.

How small can Ba be? I didn't know it some few days ago when I started thinking in this subject. That's why I asked the same question to Wilfrid Keller, to Eric Brier and to Chris Nash.

Up today I have received answer only from Chris. He conjectures that Ba may be as low as 4847*659 but not provable, or near 78557*509203 and [probably] provable…J

That was enough for me and for this puzzle!… and while someone develops a stronger argument to bound Ba, I propose to construct the following finite monotonously increasing sequence of k-values, k0, k1, k2, k3, … such that:

·        k0<k1<k2<k3, …< Ba where all ki are the lowest possible values

·        ki*2^x+1  and ki*2^x-1 are composite from x = 1 to Ni

·        or ki*2^x+1 or ki*2^x-1 is prime for x = Ni +1

·        Ni<Ni+1

The hope behind this sequence defined and proposed to construct, is one day to find/reach at the end of this sequence, which by definition must be Ba. This is why I call this sequence “a stepladder to Ba sequence".

Just to start from an interesting basis level, in the Table shown below you can find the first 16 members of the sequence that I have gotten mainly Ubasic (except for the last three entries where I needed the Yves Gallot's Proth.exe)

 ki Ni such that ki*2^n+1  and ki*2^n-1 are composite from n = 1 to Ni The first prime at Ni +1 Author/Date 13 1+ 13 *2^ 2 +1 C. Rivera, Dic/99 47 3- 47 *2^ 4 –1 Idem 59 4+ 59 *2^ 5 +1 Idem 109 5+ 109 *2^ 6 +1 Idem 241 10- 241 *2^ 11 -1 Idem 335 17- 335 *2^ 18 -1 Idem 1109 19- 1109 *2^ 20 -1 Idem 1373 27- 1373 *2^ 28 -1 Idem 1447 69+ 1447 *2^ 70 +1 Idem 14893 105+ 14893 *2^ 106 +1 Idem 52267 149+ 52267 *2^ 150 +1 Idem 56543 725- 56543 *2^ 726 -1 Idem 649603 2905+ 649603 *2^ 2906 +1 Idem 838441 7430- 838441*2^7431-1 Idem 8840101 14072- 8840101*2^14073-1 Idem 16935761 22393- 16935761*2^22394-1 Idem 100604613 41421- 100604613*2^411422-1 Wilfrid Keller (17/1/00) 118373279 82586+ 118373279*2^82587+1 Wilfrid Keller (17/1/00) 270704167 85460- 270704167*2^85461-1 Wilfrid Keller (17/1/00) 1355477231 356980+ 1355477231*2^356981 + 1 (107472 digits) Wilfrid Keller (24/4/02) The search keeps going by A team formed by Carlos Rivera, Felice Russo, Enoch Haga, Jeff Heleen & Patrick De Geest 2294020991 800492+ 2294020991*2^800493+1 P/Ba Team Ba All n ? ?

As you can observe the first consequence of this little search is that the low limit 4847*659 = 3194173 <16935761 given by Nash has been superseded, but we are still far from 78557*509203 = 40,001,460,071, so there is still a small chance that we find an easy (small) and good candidate to Ba.

How easier is this approach to hunt Ba? I don't know (or maybe I know it  but I don't want to realize it completely because this puzzle may disappears…)

Hopefully it is just as hard as any other approach, but wait a second! … Working hard to get more and more members of this sequence I just want to recall your attention that each time you fail to get Ba as a compensation you have an extra incentive-prize: you get a prime!… Yes indeed, as the sequence grows each time you fail to get Ba very soon you'll get a really big prime!!!….

An example from the Sierpinski search field:  While 78557 is still the conjectured lowest Sierpinski number, there remain 19 "probable Sierpinski numbers" (see http://www.prothsearch.net/index.html ) being the least probable Sierpinski number 4847 which remains producing composite numbers up to n= 279700. But in the meanwhile the search has produced sad&happy news like the following one:

"On March 25, 1999, the value k = 74269 was eliminated by Marc Thibeault, who found the prime 74269 .*2^167546 + 1 using Yves Gallot's program Proth.exe."

And there are other 19 sad news&happy news in the future!…

Question: Would you like to extend this stepladder to Ba  sequence?

Hint: probably we may convince to Chris Nash to develop a special version of his popular code (PrimeForm) to construct this sequence….

Solution

Wilfrid Keller has produced some new and larger results for the table in construction here (See table above) . These results are just a part of a very interesting letter sent today (17/01/2000) after the Yves Gallot's surprising new results getting smaller and smaller Brier numbers. See Problem 29  or click here to download the Keller's letter.

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One year later Wilfrid Keller is still behind a good candidate to the Ba .Please see his latest results in the Table above. In particular a promising number is k=1355477231 which produces composites up to n=200,000. Regarding this numbers Keller writes " By the way, I have been having a very close look at the divisibility patterns for both sequences, but I couldn't really discern any peculiarity -- except perhaps that many, many different "small" primes appear as the first divisor of a member in the sequence. In your pages you might also ask if someone would be able to show even for one of both sequences that it contains only composite numbers. No chance, I believe..." But...

In only a few other cases k>1355477231 remained composite for n > 50000 (to choose some arbitrary limit):

 k exponent 282681079 58782- 343689013 58313+ 637741513 78919+ 1191503821 76759+ 1438096843 59386- 1501846421 125702+

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