Problems & Puzzles: Problems Problem 36. The LiskovetsGallot numbers Valery Liskovets, while studying the "List of primes k.2^{n} + 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 Sierpinskilike shape I would say that
the Liskovets
conjectures that: All this was communicated by email by Liskovets to Yves Gallot the Thursday 24 of May, not without extending his conjectural thinking to the k*2^n1 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 LiskovetsGallot 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^n1 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. Questions: 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?
Solution 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?) *** JeanClaude 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^n1, 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 & LiskovetGallot numbers hunters. *** 





