Penta-power prime seeds: Difference between revisions
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{{
Generate the sequence of penta-power prime seeds: positive integers '''n''' such that:
Line 230:
</pre>
=={{header|F_Sharp|F#}}==
<syntaxhighlight lang="fsharp">
// Penta-power prime seeds. Nigel Galloway: April 5th., 2023
let fG n g=let n=bigint(n:int) in let n=n**g+n+1I in Open.Numeric.Primes.MillerRabin.IsProbablePrime &n
let fN(n,g)=Seq.initInfinite((+)n)|>Seq.filter(fun n->let g=fG n in g 0&&g 1&&g 2&&g 3&&g 4)|>Seq.mapi(fun n g->(n,g))|>Seq.find(snd>>(<)g)
Seq.initInfinite((*)2>>(+)1)|>Seq.filter(fun n->let g=fG n in g 0&&g 1&&g 2&&g 3&&g 4)|>Seq.take 30|>Seq.iter(printf "%d "); printfn "\n"
[1000000..1000000..10000000]|>Seq.scan(fun(n,g,x) l->let i,e=fN(g,l) in (n+i,e,l))(0,0,0)|>Seq.skip 1|>Seq.iter(fun(n,g,l)->printfn $"First element over %8d{l} is %9d{g} at index %3d{n}")
</syntaxhighlight>
{{out}}
<pre>
1 5 69 1665 2129 25739 29631 62321 77685 80535 82655 126489 207285 211091 234359 256719 366675 407945 414099 628859 644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321
First element over 1000000 is 1121189 at index 25
First element over 2000000 is 2066079 at index 38
First element over 3000000 is 3127011 at index 46
First element over 4000000 is 4059525 at index 50
First element over 5000000 is 5279175 at index 58
First element over 6000000 is 6320601 at index 62
First element over 7000000 is 7291361 at index 67
First element over 8000000 is 8334915 at index 68
First element over 9000000 is 9100671 at index 70
First element over 10000000 is 10347035 at index 71
</pre>
=={{header|Factor}}==
{{works with|Factor|0.99 2022-04-03}}
Line 253 ⟶ 276:
1,121,189 1,158,975 1,483,691 1,490,475 1,512,321
</pre>
=={{header|FreeBASIC}}==
{{libheader|GMP}}
<syntaxhighlight lang="freebasic">' version 13-04-2023
' compile with: fbc -s console
#Include "gmp.bi"
#Define sieve_max 21000000
Dim As Mpz_ptr n2 = Allocate (Len(__mpz_struct))
Dim As Mpz_ptr n3 = Allocate (Len(__mpz_struct))
Dim As Mpz_ptr n4 = Allocate (Len(__mpz_struct))
Mpz_init(n2) : Mpz_init(n3) : Mpz_init(n4)
Dim As ULongInt i, j
ReDim As boolean sieve(sieve_max)
' default value on initialization is FALSE
sieve(2) = TRUE
' set all odd numbers to TRUE
For i = 3 To sieve_max Step 2
sieve(i) = TRUE
Next
For i = 3 To Sqr(sieve_max) Step 2
If sieve(i) = TRUE Then
For j = i * i To sieve_max Step i * 2
sieve(j) = FALSE
Next
End If
Next
Dim As LongInt n = -1, count, k
Dim As LongInt si = 15
Print "The first thirty penta-power prime seeds are:"
While count < 30
n += 2
k = n +1
' n ^ 0 = 1
If sieve(1 + k) And sieve(n + k) Then ' skip if 1 + k or n + k is not prime
Mpz_ui_pow_ui(n4, n , 4)
Mpz_add_ui(n4, n4, k)
If Mpz_probab_prime_p(n4, si) < 1 Then Continue While ' skip if not prime
Mpz_ui_pow_ui(n3, n, 3)
Mpz_add_ui(n3, n3, k)
If Mpz_probab_prime_p(n3, si) < 1 Then Continue While ' skip if not prime
Mpz_ui_pow_ui(n2, n, 2)
Mpz_add_ui(n2, n2, k)
If Mpz_probab_prime_p(n2, si) >= 1 Then ' if prime then print n
Print Using "##########"; n;
count += 1
If (count Mod 10) = 0 Then Print
End If
End If
Wend
Dim As ULongInt m = 1, million = 1000000
n = -1 : count = 0
Print !"\n\nFirst penta-power prime seed greater than:"
While m < 11
n += 2
k = n +1
If sieve(1 + k) And sieve(n + k) Then ' skip if 1 + k or n + k is not prime
Mpz_ui_pow_ui(n4, n , 4)
Mpz_add_ui(n4, n4, k)
If Mpz_probab_prime_p(n4, si) < 1 Then Continue While ' skip if not prime
Mpz_ui_pow_ui(n3, n, 3)
Mpz_add_ui(n3, n3, k)
If Mpz_probab_prime_p(n3, si) < 1 Then Continue While ' skip if not prime
Mpz_ui_pow_ui(n2, n, 2)
Mpz_add_ui(n2, n2, k)
If Mpz_probab_prime_p(n2, si) >= 1 Then
count += 1
If n > million Then
Print Using " ## million is #########, at index ### "; m; n; count
m += 1
million = m * 1000000
End If
End If
End If
Wend
Mpz_clear(n4) : Mpz_clear(n3) : Mpz_clear(n2)
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End</syntaxhighlight>
{{out}}
<pre>The first thirty penta-power prime seeds are:
1 5 69 1665 2129 25739 29631 62321 77685 80535
82655 126489 207285 211091 234359 256719 366675 407945 414099 628859
644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321
First penta-power prime seed greater than:
1 million is 1,121,189 at index 26
2 million is 2,066,079 at index 39
3 million is 3,127,011 at index 47
4 million is 4,059,525 at index 51
5 million is 5,279,175 at index 59
6 million is 6,320,601 at index 63
7 million is 7,291,361 at index 68
8 million is 8,334,915 at index 69
9 million is 9,100,671 at index 71
10 million is 10,347,035 at index 72</pre>
=={{header|Go}}==
Line 364 ⟶ 495:
9 million is the 71st: 9,100,671
10 million is the 72nd: 10,347,035
</pre>
=={{header|J}}==
<syntaxhighlight lang=j> ps=. ] #~ 1 p: 1 + ^~ + ]
_10 ]\ 4x ps 3x ps 2 ps 1 ps 0 ps i. 1520000
1 5 69 1665 2129 25739 29631 62321 77685 80535
82655 126489 207285 211091 234359 256719 366675 407945 414099 628859
644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321</syntaxhighlight>
=={{header|Java}}==
<syntaxhighlight lang="java">
import java.math.BigInteger;
public final class PentaPowerPrimeSeeds {
public static void main(String[] args) {
System.out.println("The first 30 penta-power prime seeds:");
int index = 0;
int number = 1;
boolean searching = true;
while ( searching ) {
if ( isPentaPowerPrimeSeed(number) ) {
index += 1;
if ( index <= 30 ) {
System.out.print(String.format("%7d%s", number, ( index % 6 == 0 ? "\n" : " " )));
} else if ( number > 10_000_000 ) {
System.out.println();
System.out.println("The first penta-power prime seed greater than 10,000,000 is "
+ number + " at index " + index);
searching = false;
}
}
number += 2;
}
}
private static boolean isPentaPowerPrimeSeed(long number) {
BigInteger p = BigInteger.ONE;
BigInteger nPlus1 = BigInteger.valueOf(number + 1);
for ( int i = 0; i <= 4; i++ ) {
if ( ! p.add(nPlus1).isProbablePrime(15) ) {
return false;
}
p = p.multiply(BigInteger.valueOf(number));
}
return true;
}
}
</syntaxhighlight>
{{ out }}
<pre>
The first 30 penta-power prime seeds:
1 5 69 1665 2129 25739
29631 62321 77685 80535 82655 126489
207285 211091 234359 256719 366675 407945
414099 628859 644399 770531 781109 782781
923405 1121189 1158975 1483691 1490475 1512321
The first penta-power prime seed greater than 10,000,000 is 10347035 at index 72
</pre>
=={{header|Julia}}==
This solution uses Primes to determine primality.
<syntaxhighlight lang=julia>
using Primes, Printf
function ispenta(n)
all(0:4) do i
isprime(n^i + n + 1)
end
end
function firstpenta(m, T=BigInt)
nums = Iterators.countfrom(T(1))
pentas = Iterators.filter(ispenta, nums)
firstn = Iterators.take(pentas, m)
return collect(firstn)
end
function table_display(nums, num_columns)
num_elements = length(nums)
num_rows = div(num_elements, num_columns)
remaining_elements = num_elements % num_columns
for i in 1:num_rows
for j in 1:num_columns
index = (i - 1) * num_columns + j
print(nums[index], "\t")
end
println()
end
for i in 1:remaining_elements
index = num_rows * num_columns + i
print(nums[index], "\t")
end
println()
end
function stretch_penta(goal, T=BigInt)
nums = Iterators.countfrom(T(1))
pentas = Iterators.filter(ispenta, nums)
firstn = Iterators.takewhile(<=(goal), pentas)
return collect(firstn)
end
function run_rosetta()
fp = firstpenta(30)
println("First 30 Penta power prime seeds:")
table_display(fp, 10)
sp = stretch_penta(20000000)
milestones = 1000000 .* (1:10)
for milestone in milestones
index = findfirst(>(milestone), sp)
@printf "First element over %9i: %9i, index:%4i\n" milestone sp[index] index
end
end
if abspath(PROGRAM_FILE) == @__FILE__
run_rosetta()
end
</syntaxhighlight>
<pre>
First 30 Penta power prime seeds:
1 5 69 1665 2129 25739 29631 62321 77685 80535
82655 126489 207285 211091 234359 256719 366675 407945 414099 628859
644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321
First element over 1000000: 1121189, index: 26
First element over 2000000: 2066079, index: 39
First element over 3000000: 3127011, index: 47
First element over 4000000: 4059525, index: 51
First element over 5000000: 5279175, index: 59
First element over 6000000: 6320601, index: 63
First element over 7000000: 7291361, index: 68
First element over 8000000: 8334915, index: 69
First element over 9000000: 9100671, index: 71
First element over 10000000: 10347035, index: 72
</pre>
Line 468 ⟶ 743:
207285
<terminated>
</pre>
=={{header|Nim}}==
{{libheader|Nim-Integers}}
<syntaxhighlight lang=Nim>import std/[strformat, strutils]
import integers
func isPentaPowerPrimeSeeds(n: Integer): bool =
var p = newInteger(1)
var n1 = n + 1
for _ in 0..4:
if not isPrime(p + n1): return false
p *= n
result = true
const N = 10_000_000
echo "First 30 penta-power prime seeds:"
var count = 0
var n = 1
while true:
if n.isPentaPowerPrimeSeeds():
inc count
if count <= 30:
stdout.write &"{n:7}"
stdout.write if count mod 6 == 0: '\n' else: ' '
if count == 30: echo()
elif n > N:
echo &"First penta-power prime seed greater than {insertSep($N)} " &
&"is {insertSep($n)} at position {count}."
break
inc n, 2
</syntaxhighlight>
{{out}}
<pre>First 30 penta-power prime seeds:
1 5 69 1665 2129 25739
29631 62321 77685 80535 82655 126489
207285 211091 234359 256719 366675 407945
414099 628859 644399 770531 781109 782781
923405 1121189 1158975 1483691 1490475 1512321
First penta-power prime seed greater than 10_000_000 is 10_347_035 at position 72.
</pre>
Line 624 ⟶ 942:
nine million is the 71st: 9,100,671
ten million is the 72nd: 10,347,035</pre>
=={{header|RPL}}==
Directly adapted from [[Quad-power prime seeds#RPL|Quad-power prime seeds]], but faster since seeds must be odd to get <code>n<sup>0</sup> + n + 1</code> primality. However, needs to be run on an emulator to get the result in around half an hour.
{{works with|HP|49}}
« { } 1
'''WHILE''' OVER SIZE 30 < '''REPEAT'''
1 SF
0 4 '''FOR''' j
DUP j ^ OVER + 1 +
'''IF''' ISPRIME? NOT '''THEN''' 1 CF 4 'j' STO '''END'''
'''NEXT'''
'''IF''' 1 FS? '''THEN''' SWAP OVER + SWAP '''END '''
2 +
'''END'''
» '<span style="color:blue">TASK</span>' STO
{{out}}
<pre>
1:{1 5 69 1665 2129 25739 29631 62321 77685 80535 82655 126489 207285 211091 234359 256719 366675 407945 414099 628859 644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321}
</pre>
=={{header|Ruby}}==
Line 639 ⟶ 976:
644399 770531 781109 782781 923405 1121189 1158975 1483691 1490475 1512321
</pre>
=={{header|Scala}}==
{{trans|Java}}
<syntaxhighlight lang="Scala">
import scala.annotation.tailrec
import java.math.BigInteger
object PentaPowerPrimeSeeds extends App {
println("The first 30 penta-power prime seeds:")
val first30 = Stream.from(1, 2).filter(isPentaPowerPrimeSeed).take(30)
first30.zipWithIndex.foreach { case (seed, index) =>
print(f"$seed%7d${if ((index + 1) % 6 == 0) "\n" else " "}")
}
val firstAbove10M = Stream.from(1, 2).filter(isPentaPowerPrimeSeed).find(_ > 10000000)
firstAbove10M match {
case Some(seed) => println(s"\nThe first penta-power prime seed greater than 10,000,000 is $seed")
case None => println("No penta-power prime seed greater than 10,000,000 was found.")
}
def isPentaPowerPrimeSeed(number: Int): Boolean = {
val bigIntNumber = BigInteger.valueOf(number)
val bigIntNumberPlusOne = bigIntNumber.add(BigInteger.ONE)
(0 to 4).forall { i =>
bigIntNumber.pow(i).add(bigIntNumberPlusOne).isProbablePrime(15)
}
}
}
</syntaxhighlight>
{{out}}
<pre>
The first 30 penta-power prime seeds:
1 5 69 1665 2129 25739
29631 62321 77685 80535 82655 126489
207285 211091 234359 256719 366675 407945
414099 628859 644399 770531 781109 782781
923405 1121189 1158975 1483691 1490475 1512321
The first penta-power prime seed greater than 10,000,000 is 10347035
</pre>
=={{header|Wren}}==
{{libheader|Wren-gmp}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="
import "./fmt" for Fmt
| |||