Step by Step solution about How to resolve the algorithm Zsigmondy numbers step by step in the Mathematica/Wolfram Language programming language

Function Definition

ClearAll[zsigmondy]
Attributes[zsigmondy] = Listable;

zsigmondy[a_Integer, b_Integer, 1] := a - b /; a >= b;

This defines a function zsigmondy that calculates the Zsigmondy function for the given integers a and b with an exponent n. It uses the Listable attribute to allow the function to operate element-wise on lists of input values.

The function checks if a is greater than or equal to b and if so, it returns the difference between a and b. This is the base case for the recursive calculation.

Recursive Definition

zsigmondy[a_Integer, b_Integer, n_Integer] := (
   hatvanyok = a^Range[n] - b^Range[n];
  kishatvany = a^Range[n - 1] - b^Range[n - 1];
  biggestelement = Part[hatvanyok, n];
  divisors = Divisors[biggestelement];
  divisorpairs = Join @@ (Thread[{#, kishatvany}] & /@ divisors);
  coprimepairs = Select[divisorpairs, CoprimeQ[#[[1]], #[[2]]] &];
  firstelement = coprimepairs[[All, 1]];
  revesedelements = ReverseSort[firstelement];
  Commonest[revesedelements, 1]) /; a >= b

The recursive definition handles the case where n is greater than 1.

• hatvanyok is a list of the powers of a and b up to the exponent n.
• kishatvany is a list of the powers of a and b up to the exponent n-1.
• biggestelement is the last element of hatvanyok.
• divisors is a list of the divisors of biggestelement.
• divisorpairs is a list of pairs of divisors and their corresponding elements in kishatvany.
• coprimepairs is a list of divisor pairs where the first and second elements are coprime (have no common factors).
• firstelement is a list of the first elements of the coprime pairs.
• revesedelements is a list of the first elements in firstelement sorted in reverse order.
• Commonest[revesedelements, 1] returns the most common element in revesedelements.

This recursive definition essentially finds the most common divisor of a^n - b^n and a^(n-1) - b^(n-1) among all possible divisor pairs where the first and second elements are coprime.

Data Generation

data2 = MapThread[
  zsigmondy, {{2, 3, 4, 5, 6, 7, 3, 5, 7, 7}, {1, 1, 1, 1, 1, 1, 2, 
    3, 3, 5}, {Range[10], Range[10], Range[10], Range[10], Range[10],
     Range[10], Range[10], Range[10], Range[10], Range[10]}}];

This generates data for the Zsigmondy function for various pairs of integers and exponents.

data1 = List["Zsigmondy:2,1,n", "Zsigmondy:3,1,n", "Zsigmondy:4,1,n", 
  "Zsigmondy:5,1,n", "Zsigmondy:6,1,n", "Zsigmondy:7,1,n", 
  "Zsigmondy:3,2,n", "Zsigmondy:5,3,n", "Zsigmondy:7,3,n", 
  "Zsigmondy:7,5,n"];

This defines a list of labels for the different pairs of integers and exponents.

Grid Display

Grid[Transpose@{data1, data2}~
 Prepend~{"pair of numbers", "Zsigmondy numbers"}, 
Dividers -> {All, {1 -> True, 2 -> True}}]

This creates a grid display that shows the labels and the corresponding Zsigmondy numbers. The grid is divided into columns for the pairs of numbers and the Zsigmondy numbers, and rows for each pair of integers and exponent. The first and second rows of the grid contain the labels and the Zsigmondy numbers, respectively.

ClearAll[zsigmondy]
Attributes[zsigmondy] = Listable;
zsigmondy[a_Integer, b_Integer, 1] := a - b /; a >= b;
zsigmondy[a_Integer, b_Integer, n_Integer] := (
    hatvanyok = a^Range[n] - b^Range[n];
   kishatvany = a^Range[n - 1] - b^Range[n - 1];
   biggestelement = Part[hatvanyok, n];
   divisors = Divisors[biggestelement];
   divisorpairs = Join @@ (Thread[{#, kishatvany}] & /@ divisors);
   coprimepairs = Select[divisorpairs, CoprimeQ[#[[1]], #[[2]]] &];
   firstelement = coprimepairs[[All, 1]];
   revesedelements = ReverseSort[firstelement];
   Commonest[revesedelements, 1]) /; a >= b


data2 = MapThread[
   zsigmondy, {{2, 3, 4, 5, 6, 7, 3, 5, 7, 7}, {1, 1, 1, 1, 1, 1, 2, 
     3, 3, 5}, {Range[10], Range[10], Range[10], Range[10], Range[10],
      Range[10], Range[10], Range[10], Range[10], Range[10]}}];
data1 = List["Zsigmondy:2,1,n", "Zsigmondy:3,1,n", "Zsigmondy:4,1,n", 
   "Zsigmondy:5,1,n", "Zsigmondy:6,1,n", "Zsigmondy:7,1,n", 
   "Zsigmondy:3,2,n", "Zsigmondy:5,3,n", "Zsigmondy:7,3,n", 
   "Zsigmondy:7,5,n"];
Grid[Transpose@{data1, data2}~
  Prepend~{"pair of numbers", "Zsigmondy numbers"}, 
 Dividers -> {All, {1 -> True, 2 -> True}}]


pair of numbers   Zsigmondy numbers

Zsigmondy:2,1,n   {1, {3}, {7}, {5}, {31}, {1}, {127}, {17}, {73}, {11}}

Zsigmondy:3,1,n   {2, {1}, {13}, {5}, {121}, {7}, {1093}, {41}, {757}, {61}}

Zsigmondy:4,1,n   {3, {5}, {7}, {17}, {341}, {13}, {5461}, {257}, {1387}, {41}}

Zsigmondy:5,1,n   {4, {3}, {31}, {13}, {781}, {7}, {19531}, {313}, {15751}, {521}}

Zsigmondy:6,1,n   {5, {7}, {43}, {37}, {311}, {31}, {55987}, {1297}, {46873}, {1111}}

Zsigmondy:7,1,n   {6, {1}, {19}, {25}, {2801}, {43}, {137257}, {1201}, {39331}, {2101}}

Zsigmondy:3,2,n   {1, {5}, {19}, {13}, {211}, {7}, {2059}, {97}, {1009}, {11}}

Zsigmondy:5,3,n   {2, {1}, {49}, {17}, {1441}, {19}, {37969}, {353}, {19729}, {421}}

Zsigmondy:7,3,n   {4, {5}, {79}, {29}, {4141}, {37}, {205339}, {1241}, {127639}, {341}}

Zsigmondy:7,5,n   {2, {3}, {109}, {37}, {6841}, {13}, {372709}, {1513}, {176149}, {1661}}