Zsigmondy numbers
Zsigmondy numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Zsigmondy numbers n to a, b, are the greatest divisor of an - bn that is coprime to am - bm for all positive integers m < n.
- E.G.
Suppose we set a = 2 and b = 1. (Zs(n,2,1))
For each n, find the divisors of an - bn and return the largest that is coprime to all of am - bm, where m is each of the positive integers 1 to n - 1.
When n = 4, 24 - 14 = 15. The divisors of 15 are 1, 3, 5, and 15.
For m = 1, 2, 3 we get 2-1, 22-12, 23-13, or 1, 3, 7.
The divisors of 15 that are coprime to each are 5 and 1, (1 is always included).
The largest coprime divisor is 5, so Zs(4,2,1) = 5.
When n = 6, 26 - 16 = 63; its divisors are 1, 3, 7, 9, 21, 63. The largest divisor coprime to all of 1, 3, 7, 15, 31 is 1, (1 is always included), so Zs(6,2,1) = 1.
If a particular an - bn is prime, then Zs(n,a,b) will be equal to that prime. 25 - 15 = 31 so Zs(5,2,1) = 31.
- Task
- Write a general routine (function, procedure, whatever) to find the Zsigmondy number sequence given a set of radices.
- Use that routine to generate the first several elements, (at least 10), for the following radix sets.
- (2,1)
- (3,1)
- (4,1)
- (5,1)
- (6,1)
- (7,1)
- (3,2)
- (5,3)
- (7,3)
- (7,5)
- See also
- OEIS:A064078 - Zsigmondy numbers for a = 2, b = 1
- OEIS:A064077 - Zsigmondy numbers for a = 3, b = 1
- OEIS:A064080 - Zsigmondy numbers for a = 4, b = 1
- OEIS:A064081 - Zsigmondy numbers for a = 5, b = 1
- OEIS:A064082 - Zsigmondy numbers for a = 6, b = 1
- OEIS:A064083 - Zsigmondy numbers for a = 7, b = 1
- OEIS:A109325 - Zsigmondy numbers for a = 3, b = 2
- OEIS:A109347 - Zsigmondy numbers for a = 5, b = 3
- OEIS:A109348 - Zsigmondy numbers for a = 7, b = 3
- OEIS:A109349 - Zsigmondy numbers for a = 7, b = 5
J
Implementation:
dp=: -/@:(^/) NB. (a,b) dp n is (a^n)-(b^n)
Zsigmondy=: dp (-.,)&.:q: (dp 1 }. i.)
In other words, 2 1 dp 1 2 3 4 is 1 3 7 15. And coprime is sequence difference (not set difference, since prime factors may repeat) under factorization (generate the sequence of prime factors for each number, remove any common factors and form the product of any that remain).
Task examples:
Zs=: Zsigmondy"1 0&(1x+i.20) NB. first 20 in sequence
Zs 2 1
1 3 7 5 31 1 127 17 73 11 2047 13 8191 43 151 257 131071 19 524287 41
Zs 3 1
2 1 13 5 121 7 1093 41 757 61 88573 73 797161 547 4561 3281 64570081 703 581130733 1181
Zs 4 1
3 5 7 17 341 13 5461 257 1387 41 1398101 241 22369621 3277 49981 65537 5726623061 4033 91625968981 61681
Zs 5 1
4 3 31 13 781 7 19531 313 15751 521 12207031 601 305175781 13021 315121 195313 190734863281 5167 4768371582031 375601
Zs 6 1
5 7 43 37 311 31 55987 1297 46873 1111 72559411 1261 2612138803 5713 1406371 1679617 3385331888947 46441 121871948002099 1634221
Zs 7 1
6 1 19 25 2801 43 137257 1201 39331 2101 329554457 2353 16148168401 102943 4956001 2882401 38771752331201 117307 1899815864228857 1129901
Zs 3 2
1 5 19 13 211 7 2059 97 1009 11 175099 61 1586131 463 3571 6817 129009091 577 1161737179 4621
Zs 5 3
2 1 49 17 1441 19 37969 353 19729 421 24325489 481 609554401 10039 216001 198593 381405156481 12979 9536162033329 288961
Zs 7 3
4 5 79 29 4141 37 205339 1241 127639 341 494287399 2041 24221854021 82573 3628081 2885681 58157596211761 109117 2849723505777919 4871281
Zs 7 5
2 3 109 37 6841 13 372709 1513 176149 1661 964249309 1801 47834153641 75139 3162961 3077713 115933787267041 30133 5689910849522509 3949201
Julia
""" Rosetta code task: rosettacode.org/wiki/Zsigmondy_numbers """
using Primes
function divisors(n)
f = [one(n)]
for (p,e) in factor(n)
f = reduce(vcat, [f*p^j for j in 1:e], init=f)
end
return length(f) == 1 ? [one(n), n] : sort!(f)
end
function Zs(n, a, b)
@assert a > b
dexpms = map(i -> a^i - b^i, 1:n-1)
dexpn = a^n - b^n
return maximum(reduce(vcat, filter(d -> all(k -> gcd(k, d) == 1, dexpms), divisors(dexpn))))
end
abs = [(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (3, 2), (5, 3), (7, 3), (7, 5)]
for (a, b) in abs
println("\nZsigmondy(n, $a, $b): ", join([Zs(n, a, b) for n in 1:20], ", "))
end
- Output:
Zsigmondy(n, 2, 1): 1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41 Zsigmondy(n, 3, 1): 2, 1, 13, 5, 121, 7, 1093, 41, 757, 61, 88573, 73, 797161, 547, 4561, 3281, 64570081, 703, 581130733, 1181 Zsigmondy(n, 4, 1): 3, 5, 7, 17, 341, 13, 5461, 257, 1387, 41, 1398101, 241, 22369621, 3277, 49981, 65537, 5726623061, 4033, 91625968981, 61681 Zsigmondy(n, 5, 1): 4, 3, 31, 13, 781, 7, 19531, 313, 15751, 521, 12207031, 601, 305175781, 13021, 315121, 195313, 190734863281, 5167, 4768371582031, 375601 Zsigmondy(n, 6, 1): 5, 7, 43, 37, 311, 31, 55987, 1297, 46873, 1111, 72559411, 1261, 2612138803, 5713, 1406371, 1679617, 3385331888947, 46441, 121871948002099, 1634221 Zsigmondy(n, 7, 1): 6, 1, 19, 25, 2801, 43, 137257, 1201, 39331, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 2882401, 38771752331201, 117307, 1899815864228857, 1129901 Zsigmondy(n, 3, 2): 1, 5, 19, 13, 211, 7, 2059, 97, 1009, 11, 175099, 61, 1586131, 463, 3571, 6817, 129009091, 577, 1161737179, 4621 Zsigmondy(n, 5, 3): 2, 1, 49, 17, 1441, 19, 37969, 353, 19729, 421, 24325489, 481, 609554401, 10039, 216001, 198593, 381405156481, 12979, 9536162033329, 288961 Zsigmondy(n, 7, 3): 4, 5, 79, 29, 4141, 37, 205339, 1241, 127639, 341, 494287399, 2041, 24221854021, 82573, 3628081, 2885681, 58157596211761, 109117, 2849723505777919, 4871281 Zsigmondy(n, 7, 5): 2, 3, 109, 37, 6841, 13, 372709, 1513, 176149, 1661, 964249309, 1801, 47834153641, 75139, 3162961, 3077713, 115933787267041, 30133, 5689910849522509, 3949201
Raku
First twenty elements of each.
use Prime::Factor;
sub Zsigmondy ($a, $b) {
my @aexp = 1, $a, * × $a … *;
my @bexp = 1, $b, * × $b … *;
(1..∞).map: -> $n {
my @divisors = divisors(@aexp[$n] - @bexp[$n]).sort: -*;
@divisors.first: -> $d { all (1..^$n).map: -> $m { (@aexp[$m] - @bexp[$m]) gcd $d == 1 } }
}
}
for 'A064078: Zsigmondy(n,2,1)', (2,1),
'A064079: Zsigmondy(n,3,1)', (3,1),
'A064080: Zsigmondy(n,4,1)', (4,1),
'A064081: Zsigmondy(n,5,1)', (5,1),
'A064082: Zsigmondy(n,6,1)', (6,1),
'A064083: Zsigmondy(n,7,1)', (7,1),
'A109325: Zsigmondy(n,3,2)', (3,2),
'A109347: Zsigmondy(n,5,3)', (5,3),
'A109348: Zsigmondy(n,7,3)', (7,3),
'A109349: Zsigmondy(n,7,5)', (7,5)
-> $name, $seq { say "\n$name:\n" ~ Zsigmondy(|$seq)[^20] }
- Output:
A064078: Zsigmondy(n,2,1): 1 3 7 5 31 1 127 17 73 11 2047 13 8191 43 151 257 131071 19 524287 41 A064079: Zsigmondy(n,3,1): 2 1 13 5 121 7 1093 41 757 61 88573 73 797161 547 4561 3281 64570081 703 581130733 1181 A064080: Zsigmondy(n,4,1): 3 5 7 17 341 13 5461 257 1387 41 1398101 241 22369621 3277 49981 65537 5726623061 4033 91625968981 61681 A064081: Zsigmondy(n,5,1): 4 3 31 13 781 7 19531 313 15751 521 12207031 601 305175781 13021 315121 195313 190734863281 5167 4768371582031 375601 A064082: Zsigmondy(n,6,1): 5 7 43 37 311 31 55987 1297 46873 1111 72559411 1261 2612138803 5713 1406371 1679617 3385331888947 46441 121871948002099 1634221 A064083: Zsigmondy(n,7,1): 6 1 19 25 2801 43 137257 1201 39331 2101 329554457 2353 16148168401 102943 4956001 2882401 38771752331201 117307 1899815864228857 1129901 A109325: Zsigmondy(n,3,2): 1 5 19 13 211 7 2059 97 1009 11 175099 61 1586131 463 3571 6817 129009091 577 1161737179 4621 A109347: Zsigmondy(n,5,3): 2 1 49 17 1441 19 37969 353 19729 421 24325489 481 609554401 10039 216001 198593 381405156481 12979 9536162033329 288961 A109348: Zsigmondy(n,7,3): 4 5 79 29 4141 37 205339 1241 127639 341 494287399 2041 24221854021 82573 3628081 2885681 58157596211761 109117 2849723505777919 4871281 A109349: Zsigmondy(n,7,5): 2 3 109 37 6841 13 372709 1513 176149 1661 964249309 1801 47834153641 75139 3162961 3077713 115933787267041 30133 5689910849522509 3949201
Wren
Shows the first 20 terms for each radix set apart from [7, 1] and [7, 5] where I've had to limit the number of terms to 18 for now. This is because the 19th term is being calculated incorrectly - probably due to intermediate calculations exceeding Wren's safe integer limit of 2^53, though I'll need to investigate further to find the exact cause.
import "./math" for Int
import "./fmt" for Fmt
var zs = Fn.new { |n, a, b|
var dn = a.pow(n) - b.pow(n)
if (Int.isPrime(dn)) return dn
var divs = Int.divisors(dn)
var dms = (1...n).map { |m| a.pow(m) - b.pow(m) }.toList
for (i in divs.count-1..1) {
if (dms.all { |dm| Int.gcd(dm, divs[i]) == 1 }) return divs[i]
}
return 1
}
var abs = [ [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [3, 2], [5, 3], [7, 3], [7, 5] ]
for (ab in abs) {
var a = ab[0]
var b = ab[1]
var lim = 20
if (a == 7 && b != 3) lim = 18
System.print("Zsigmony(n, %(a), %(b)) - first %(lim) terms:")
Fmt.print("$d", (1..lim).map { |n| zs.call(n, a, b) }.toList)
System.print()
}- Output:
Zsigmony(n, 2, 1) - first 20 terms: 1 3 7 5 31 1 127 17 73 11 2047 13 8191 43 151 257 131071 19 524287 41 Zsigmony(n, 3, 1) - first 20 terms: 2 1 13 5 121 7 1093 41 757 61 88573 73 797161 547 4561 3281 64570081 703 581130733 1181 Zsigmony(n, 4, 1) - first 20 terms: 3 5 7 17 341 13 5461 257 1387 41 1398101 241 22369621 3277 49981 65537 5726623061 4033 91625968981 61681 Zsigmony(n, 5, 1) - first 20 terms: 4 3 31 13 781 7 19531 313 15751 521 12207031 601 305175781 13021 315121 195313 190734863281 5167 4768371582031 375601 Zsigmony(n, 6, 1) - first 20 terms: 5 7 43 37 311 31 55987 1297 46873 1111 72559411 1261 2612138803 5713 1406371 1679617 3385331888947 46441 121871948002099 1634221 Zsigmony(n, 7, 1) - first 18 terms: 6 1 19 25 2801 43 137257 1201 39331 2101 329554457 2353 16148168401 102943 4956001 2882401 38771752331201 117307 Zsigmony(n, 3, 2) - first 20 terms: 1 5 19 13 211 7 2059 97 1009 11 175099 61 1586131 463 3571 6817 129009091 577 1161737179 4621 Zsigmony(n, 5, 3) - first 20 terms: 2 1 49 17 1441 19 37969 353 19729 421 24325489 481 609554401 10039 216001 198593 381405156481 12979 9536162033329 288961 Zsigmony(n, 7, 3) - first 20 terms: 4 5 79 29 4141 37 205339 1241 127639 341 494287399 2041 24221854021 82573 3628081 2885681 58157596211761 109117 2849723505777919 4871281 Zsigmony(n, 7, 5) - first 18 terms: 2 3 109 37 6841 13 372709 1513 176149 1661 964249309 1801 47834153641 75139 3162961 3077713 115933787267041 30133