Problems & Puzzles: Problems
Problem 36. The Liskovets-Gallot numbers
Valery Liskovets, while studying the "List of primes k.2n + 1 for k < 300, compiled by Ray Ballinger and Wilfrid Keller" made the following penetrating observation:
As a matter of fact, counting in that list the quantity of odd or even n values for each of the k values pointed out by Valery, such that for these n values k*2^n+1 is prime, it happens that:
From this very elemental data Liskovets switched to the following intrepid conjecture:
In a more Sierpinski-like shape I would say that
All this was communicated by e-mail by Liskovets to Yves Gallot the Thursday 24 of May, not without extending his conjectural thinking to the k*2^n-1 numbers:
The same day a few hours later Yves wrote: "...(I have) applied some classical methods used to find some solutions to the Sierpinski, Riesel, Brier problem. See my note for details. I modified a little bit my programs and obtained some solutions."
These solutions (and the very first Liskovets-Gallot numbers ever produced) are:
As a matter of fact Yves says that "I conjecture that 66741, 95283, 39939, ... and 172677 are the smallest solutions for the forms - having no algebraic factorization (such as 4*2^n-1 or 9*2^-1) - but I can't prove it."
All this was communicated then by Gallot via email the same day 24 to Ray Ballinger, Wilfrid Keller and me.
In my case I decided to make the current problem presentation for the Saturday 26 issue of my site.
1. Can you confirm/verify the Gallot's first solutions
2.- Are the Yves k solutions the minimal provable ones for every case (+even, +odd, -even, -odd)
3. Find 5 more solutions for every case
4.- Are there solutions for the combined conditions +/- even or +/- odd? (this could emulate the Brier numbers). If not, can you explain why?
Today (1/6/01) Valery Liskovets wrote the following about the existence of these k values biased to even or odd n values such that k*2^n+1 is prime:
He also makes another question:
So, we have two more questions - other than the first four - and none answer. So this is getting great! (isn't it?)
Jean-Claude Rosa made a simple demonstration that 666741*2^n+1 is always composite for n even:
Rosa also sent later a similar proof than the previous one for the numbers 399939*2^n-1, n= even, but after knowing from Yves the complete set of primes that divides the numbers. He did the same for 95283*2^n+1, n=odd.
Rosa also wrote (11/06/01) the following:
He is rigorously right, but these are not "primitive" solutions, as I should demand in my question 3.
Yves Gallot has wrote in his own site a note about his search for Brier numbers that maybe will be of interest to Brier & Liskovet-Gallot numbers hunters.