Prime Puzzles · Number Theory
Calculador 3NCS
Given a pair of data N, CS, this tool generates their respective M and P values, plus 3 groups of integers that share the same CS: a) an infinite series by formula, b) a finite Keller cycle by recurrence, c) and the equivalent S/R pair.
Input data
Infinite series by formula
Nⱼ = N + j·P, for j = 2, 4, 6, ..., 18 (includes initial N, only 10 values are printed)
Keller finite cycle
N → (2·N + P) mod (2·P) ; M distinct initial numbers, then the recursion repeats
Minimum value shown in green; maximum value shown in red
S ↔ R Equivalence
R = 2·P − S (or S = 2·P − R)
Explanation
This second Calculator 3NCS.html starts from only a pair of data: N and its CS, and preliminarily calculates its M and P and then three groups of numbers: a) Only 10 numbers from an infinite sequence b) The Keller Cycle, which is a finite "cyclical" sequence, calculated by a recurrence formula c) The Conversion of N into its "opposite" N'
N can be a Sierpinski, Riesel, or Brier number, but it is not necessary to declare its type.
All the numbers calculated in the three sections share the same CS as the fed N.
The CS fed as a second piece of data is the responsibility of the user that it actually belongs to the fed N. This Calculator does NOT verify it, but takes it for granted. If the user does not know or has doubts about the CS of the N entered, he can check them in the calculator parallel to it, https://www.primepuzzles.net/CS-CalculadorV7.html.
When N is a Sierpinski or Riesel number, in the third results section, N' is simply its "opposite", Riesel or Sierpinski.
The case of Brier's numbers:
When the number N fed is Brier's, the CS to be fed is the one that results from the mixture of all the primes that are held of that Brier such as Sierpinski and Riesel, without repeating any of them.
Example: Let 38410495110832829780407 a Brier number. As Sierpinski has the CS = {3,7,11,13,19,31,37,41,61,73,151,331,1321} As Riesel has the CS = {3,5,17,97,241,257,673} So the global CS of this Brier is {3,7,11,13,19,31,37,41,61,73,151,331,1321,5,17,97,241,257,673}, which as you can see, in this case only the prime "3" was repeated and therefore only put once in the Global CS.
Feeding this Calculator 3NCS.html the following: N = 38410495110832829780407 (B) CS = {3,7,11,13,19,31,37,41,61,73,151,331,1321,5,17,97,241,257,673} The preliminary results obtained are: M = 720, P = 271126864553467513399447693469415
Then all the results of sections 1, 2 and 3 are displayed. In section 2, the Keller cycle has 720 distinct Brier numbers that share the CS of N. In section 3 N and N' are shown, that is, in this case B and B': 38410495110832829780407, 542253729068524531688062557158423
The second (B') is the result of applying the 2P-B formula and is nothing more than another Brier, with the same overall CS as B.
This second Brier, B', has its own infinite expansion and its own Keller Cycle.
Some people claim that B and B' should be considered as two members of a single cycle of integers composed of the two cycles, that is, a cycle composed by 2x720 memmbers. But I warn you that these are two completely different Cycles, which do NOT share a single integer, so -for me- it is a bit forced to call these two cycles, "a cycle".