# Penta-power prime seeds

Penta-power prime seeds 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.

Generate the sequence of penta-power prime seeds: positive integers n such that:

```   n0 + n + 1, n1 + n + 1, n2 + n + 1, n3 + n + 1 and n4 + n + 1 are all prime.
```

Task
• Find and display the first thirty penta-power prime seeds. (Or as many as are reasonably supported by your languages math capability if it is less.)

Stretch
• Find and display the position and value of first with a value greater than ten million.

See also

I can find no mention or record of this sequence anywhere. Perhaps I've invented it.

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 3.0.3 under Windows

This uses ALGOL 68G's LONG LONG INT during the Miller Rabin testing, under ALGOL 68G version three, the default precision of LONG LONG INT is 72 digits and LONG INT is 128 bit.
A sieve is used to (hopefully) quickly eliminate non-prime n+2 and 2n+1 numbers - Miller Rabin is used for n^2+n+1 etc. that are larger than the sieve.
This is about 10 times faster than the equivalent Quad-powwr prime seed program, possibly because even numbers are excluded and the n+2 test eliminates more numbers before the higher powers must be calculated..

NB: The source of the ALGOL 68-primes library is on a Rosetta Code code page linked from the above.
Note that to run this with ALGOL 68G under Windows (and probably Linux) a large heap size must be specified on the command line, e.g. `-heap 1024M`.

```BEGIN # find some Penta power prime seeds, numbers n such that:               #
#      n^p + n + 1 is prime for p = 0. 1, 2, 3, 4                       #
PR read "primes.incl.a68" PR # include prime utilities                    #
INT max prime         = 22 000 000;
# sieve the primes to max prime - 22 000 000 should be enough to allow    #
# checking primality of 2n+1 up to a little under 11 000 000 which should #
# be enough for the task                                                  #
[ 0 : max prime ]BOOL prime;
FOR i FROM LWB prime TO UPB prime DO prime[ i ] := FALSE OD;
prime[ 0 ] := prime[ 1 ] := FALSE;
prime[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO UPB prime DO prime[ i ] := TRUE  OD;
FOR i FROM 4 BY 2 TO UPB prime DO prime[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( UPB prime ) DO
IF prime[ i ] THEN
FOR s FROM i * i BY i + i TO UPB prime DO prime[ s ] := FALSE OD
FI
OD;
# returns TRUE if p is (probably) prime, FALSE otherwise                  #
#         uses the sieve if possible, Miller Rabin otherwise              #
PROC is prime = ( LONG INT p )BOOL:
IF p <= max prime
THEN prime[ SHORTEN p ]
ELSE is probably prime( p )
FI;
# attempt to find the numbers, note n^0 + n + 1 = n + 2 so n must be odd  #
BOOL finished       := FALSE;
INT  count          := 0;
INT  next limit     := 1 000 000;
INT  limit increment = next limit;
[ 10 ]INT first, limit, index;
INT  l count        := 0;
print( ( "First 30 Penta power prime seeds:", newline ) );
FOR n BY 2 WHILE NOT finished DO
IF prime[ n + 2 ] THEN
IF  INT n1 = n + 1;
prime[ n + n1 ]
THEN
# n^0 + n + 1 and n^1 + n + 1 are prime                       #
LONG INT np := LENG n * LENG n;
IF is prime( np + n1 ) THEN
# n^2 + n + 1 is prime                                    #
IF is prime( ( np *:= n ) + n1 ) THEN
# n^3 + n + 1 is prime                                #
IF is prime( ( np * n ) + n1 ) THEN
# n^4 + n + 1 is prime - have a suitable number   #
count +:= 1;
IF n > next limit THEN
# found the first element over the next limit #
first[ l count +:= 1 ] := n;
limit[ l count       ] := next limit;
index[ l count       ] := count;
next limit            +:= limit increment;
finished               := l count >= UPB first
FI;
IF count <= 30 THEN
# found one of the first 30 numbers           #
print( ( " ", whole( n, -8 ) ) );
IF count MOD 10 = 0 THEN print( ( newline ) ) FI
FI
FI
FI
FI
FI
FI
OD;
print( ( newline ) );
FOR i TO UPB first DO
print( ( "First element over ", whole( limit[ i ], -10 )
, ": ",                  whole( first[ i ], -10 )
, ", index: ",           whole( index[ i ],  -6 )
, newline
)
)
OD
END```
Output:
```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
```

## Factor

Works with: Factor version 0.99 2022-04-03
```USING: grouping io kernel lists lists.lazy math math.functions
math.primes prettyprint tools.memory.private ;

: seed? ( n -- ? )
5 [ dupd ^ 1 + + prime? ] with all-integers? ;

: pentas ( -- list )
1 lfrom [ seed? ] lfilter [ commas ] lmap-lazy ;

"First thirty penta-power prime seeds:" print
30 pentas ltake list>array 5 group simple-table.
```
Output:
```First thirty penta-power prime seeds:
1         5         69        1,665     2,129
25,739    29,631    62,321    77,685    80,535
82,655    126,489   207,285   211,091   234,359
256,719   366,675   407,945   414,099   628,859
644,399   770,531   781,109   782,781   923,405
1,121,189 1,158,975 1,483,691 1,490,475 1,512,321
```

## jq

The specified tasks are beyond the capabilties of the current implementations of jq:

• The C implementation (jq) does not support unbounded-precision integer arithmetic, and so is in effect only capable of generating the first few penta-power prime (ppp) seeds.
• The Go implementation (gojq) does support unbounded-precision integer arithmetic and the following program has been used to generate the first 13 ppp seeds, but gojq takes a long while to do so; other variants of the program have been considered but they run into gojq's memory-management limitations.

The program given below may nevertheless be of some interest as it illustrates how a JSON dictionary can be used to cache the primes up to a certain limit, and how this can be done using a sieve economically:

```# Create a dictionary for the primes up to .
def primeDictionary:
# The array we use for the sieve only stores information for the odd integers greater than 1:
#  index   integer
#      0         3
#      k   2*k + 3
# So if we wish to mark m = 2*k + 3, the relevant index is: m - 3 / 2
def ix:
if . % 2 == 0 then null
else ((. - 3) / 2)
end;

# erase(i) sets .[i*j] to false for odd integral j > i, and assumes i is odd
def erase(i):
((i - 3) / 2) as \$k
# Consider relevant multiples:
| (((length * 2 + 3) / i)) as \$upper
# ... only consider odd multiples from i onwards
| reduce range(i; \$upper; 2) as \$j (.;
(((i * \$j) - 3) / 2) as \$m
| if .[\$m] then .[\$m] = false else . end);

if . < 2 then {}
else (. + 1) as \$n
| ((\$n|sqrt) / 2) as \$s
| [range(3; \$n; 2)|true]
| reduce (1 + (2 * range(1; \$s)) ) as \$i (.; erase(\$i))
| . as \$sieve
| reduce (2, (range(3; \$n; 2) | select(\$sieve[ix]))) as \$i ({}; .[\$i|tostring]=\$i)
end ;```

```# Input should be an integer
def isPrime:
. as \$n
| if    (\$n < 2)      then false
elif (\$n % 2 == 0)  then \$n == 2
elif (\$n % 3 == 0)  then \$n == 3
else 5
| until( . <= 0;
if .*. > \$n then -1
elif (\$n % . == 0) then 0
else . + 2
|  if (\$n % . == 0) then 0
else . + 4
end
end)
| . == -1
end;

# \$primedictionary should be a dictionary of primes up to \$small
def ispentapowerprime(\$primedictionary; \$small):

def isp: if . <= \$small then \$primedictionary[tostring] else isPrime end;

. as \$n
| (. * .) as \$n2
| (. * \$n2) as \$n3
| all(\$n + 2, \$n + \$n + 1, \$n2 + \$n + 1, \$n3 + \$n + 1, \$n3 * \$n + \$n + 1; isp);

# Output: a stream of the first \$count penta-power prime-seeds
# The size of the dictionary has been chosen with gojq in mind.
def ppprimes(\$count):
# The size of primeDictionary has been chosen with gojq's limitations in mind
(\$count | .*.*. | primeDictionary) as \$pd

| limit(\$count; 1, 2, range(3; infinite; 2) | select(ispentapowerprime(\$pd; \$small)) );

ppprimes(30)```
Output:

Using gojq, progress effectively grinds to a halt after 207285.

```1
5
69
1665
2129
25739
29631
62321
77685
80535
82655
126489
207285
<terminated>
```

## Perl

Library: ntheory
```use v5.36;
use bigint;
use ntheory 'is_prime';
use List::Util 'max';

sub comma { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r }
sub table (\$c, @V) { my \$t = \$c * (my \$w = 2 + max map {length} @V); ( sprintf( ('%'.\$w.'s')x@V, @V) ) =~ s/.{1,\$t}\K/\n/gr }

my(\$i,@ppps);
while (@ppps < 30) {
my \$k = 1 + (my \$n = 2 * \$i++ + 1);
push @ppps, comma \$n if
is_prime( 1    + \$k) and
is_prime(\$n    + \$k) and
is_prime(\$n**2 + \$k) and
is_prime(\$n**3 + \$k) and
is_prime(\$n**4 + \$k);
}

say 'First thirty penta-power prime seeds:';
say table(10,@ppps);
```
Output:
```First thirty penta-power prime seeds:
1          5         69      1,665      2,129     25,739     29,631     62,321     77,685     80,535
82,655    126,489    207,285    211,091    234,359    256,719    366,675    407,945    414,099    628,859
644,399    770,531    781,109    782,781    923,405  1,121,189  1,158,975  1,483,691  1,490,475  1,512,321```

## Phix

```with javascript_semantics
include mpfr.e
mpz {p,q} = mpz_inits(2)

function isPentaPowerPrimeSeed(integer n)
-- (we already know n+2 is prime)
mpz_set_si(p,n)
for i=1 to 4 do
if i>1 then mpz_mul_si(p,p,n) end if
mpz_add_ui(q,p,n+1)
if not mpz_prime(q) then return false end if
end for
return true
end function

sequence ppps = {}
integer pn = 1, m = 1, c = 0
string l = ""
while m<=10 do
integer n = get_prime(pn)-2
if isPentaPowerPrimeSeed(n) then
c += 1
if c<31 then
ppps &= n
if c=30 then
printf(1,"First thirty penta-power prime seeds:\n%s\n",
{join_by(ppps,1,10," ",fmt:="%,9d")})
end if
end if
if n > m * 1e6 then
l &= sprintf(" %5s million is %,d (the %s)\n",
{ordinal(m,true), n, ordinal(c)})
m += 1
end if
end if
pn += 1
end while
printf(1,"First penta-power prime seed greater than:\n%s",l)
```
Output:
```First thirty penta-power prime seeds:
1         5        69     1,665     2,129    25,739    29,631    62,321    77,685    80,535
82,655   126,489   207,285   211,091   234,359   256,719   366,675   407,945   414,099   628,859
644,399   770,531   781,109   782,781   923,405 1,121,189 1,158,975 1,483,691 1,490,475 1,512,321

First penta-power prime seed greater than:
one million is 1,121,189 (the twenty-sixth)
two million is 2,066,079 (the thirty-ninth)
three million is 3,127,011 (the forty-seventh)
four million is 4,059,525 (the fifty-first)
five million is 5,279,175 (the fifty-ninth)
six million is 6,320,601 (the sixty-third)
seven million is 7,291,361 (the sixty-eighth)
eight million is 8,334,915 (the sixty-ninth)
nine million is 9,100,671 (the seventy-first)
ten million is 10,347,035 (the seventy-second)
```

## Python

```from sympy import isprime

def ispentapowerprime(n):
return all(isprime(i) for i in [n + 2, n + n + 1, n**2 + n + 1, n**3 + n + 1, n**4 + n + 1])

ppprimes = [i for i in range(10_400_000) if ispentapowerprime(i)]

for i in range(50):
print(f'{ppprimes[i]: 11,}', end='\n' if (i + 1) % 10 == 0 else '')

for n in range(1_000_000, 10_000_001, 1_000_000):
proot = next(filter(lambda x: x > n, ppprimes))
print(f'The first penta-power prime seed over {n:,} is {proot:,}')
```
Output:
```          1          5         69      1,665      2,129     25,739     29,631     62,321     77,685     80,535
82,655    126,489    207,285    211,091    234,359    256,719    366,675    407,945    414,099    628,859
644,399    770,531    781,109    782,781    923,405  1,121,189  1,158,975  1,483,691  1,490,475  1,512,321
1,711,991  1,716,989  1,780,485  1,791,041  1,835,589  1,860,011  1,861,259  1,980,441  2,066,079  2,211,705
2,215,529  2,271,009  2,413,265  2,514,161  2,915,109  2,940,405  3,127,011  3,587,319  3,890,769  3,992,379
The first penta-power prime seed over 1,000,000 is 1,121,189
The first penta-power prime seed over 2,000,000 is 2,066,079
The first penta-power prime seed over 3,000,000 is 3,127,011
The first penta-power prime seed over 4,000,000 is 4,059,525
The first penta-power prime seed over 5,000,000 is 5,279,175
The first penta-power prime seed over 6,000,000 is 6,320,601
The first penta-power prime seed over 7,000,000 is 7,291,361
The first penta-power prime seed over 8,000,000 is 8,334,915
The first penta-power prime seed over 9,000,000 is 9,100,671
The first penta-power prime seed over 10,000,000 is 10,347,035
```

## Raku

```use Lingua::EN::Numbers;

my @ppps = lazy (^∞).hyper(:5000batch).map(* × 2 + 1).grep: -> \n { my \k = n + 1; (1+k).is-prime && (n+k).is-prime && (n²+k).is-prime && (n³+k).is-prime && (n⁴+k).is-prime }

say "First thirty penta-power prime seeds:\n" ~ @ppps[^30].batch(10)».&comma».fmt("%9s").join: "\n";

say "\nFirst penta-power prime seed greater than:";

for 1..10 {
my \$threshold = Int(1e6 × \$_);
my \$key = @ppps.first: * > \$threshold, :k;
say "{\$threshold.&cardinal.fmt: '%13s'} is the {ordinal-digit \$key + 1}: {@ppps[\$key].&comma}";
}
```
Output:
```First thirty penta-power prime seeds:
1         5        69     1,665     2,129    25,739    29,631    62,321    77,685    80,535
82,655   126,489   207,285   211,091   234,359   256,719   366,675   407,945   414,099   628,859
644,399   770,531   781,109   782,781   923,405 1,121,189 1,158,975 1,483,691 1,490,475 1,512,321

First penta-power prime seed greater than:
one million is the 26th: 1,121,189
two million is the 39th: 2,066,079
three million is the 47th: 3,127,011
four million is the 51st: 4,059,525
five million is the 59th: 5,279,175
six million is the 63rd: 6,320,601
seven million is the 68th: 7,291,361
eight million is the 69th: 8,334,915
nine million is the 71st: 9,100,671
ten million is the 72nd: 10,347,035```

## Wren

Library: Wren-gmp
Library: Wren-fmt
```import "./gmp" for Mpz
import "./fmt" for Fmt

var p = Mpz.new()
var q = Mpz.one

var isPentaPowerPrimeSeed = Fn.new { |n|
p.setUi(n)
var k = n + 1
return (q + k).probPrime(15) > 0 &&
(p + k).probPrime(15) > 0 &&
(p.mul(n) + k).probPrime(15) > 0 &&
(p.mul(n) + k).probPrime(15) > 0 &&
(p.mul(n) + k).probPrime(15) > 0
}

var ppps = []
var n = 1
while (ppps.count < 30) {
if (isPentaPowerPrimeSeed.call(n)) ppps.add(n)
n = n + 2  // n must be odd
}
System.print("First thirty penta-power prime seeds:")
Fmt.tprint("\$,9d", ppps, 10)

System.print("\nFirst penta-power prime seed greater than:")
n = 1
var m = 1
var c = 0
while (true) {
if (isPentaPowerPrimeSeed.call(n)) {
c = c + 1
if (n > m * 1e6) {
Fmt.print(" \$2d million is the \$r: \$,10d", m, c, n)
m = m + 1
if (m == 11) return
}
}
n = n + 2
}```
Output:
```First thirty penta-power prime seeds:
1         5        69     1,665     2,129    25,739    29,631    62,321    77,685    80,535
82,655   126,489   207,285   211,091   234,359   256,719   366,675   407,945   414,099   628,859
644,399   770,531   781,109   782,781   923,405 1,121,189 1,158,975 1,483,691 1,490,475 1,512,321

First penta-power prime seed greater than:
1 million is the 26th:  1,121,189
2 million is the 39th:  2,066,079
3 million is the 47th:  3,127,011
4 million is the 51st:  4,059,525
5 million is the 59th:  5,279,175
6 million is the 63rd:  6,320,601
7 million is the 68th:  7,291,361
8 million is the 69th:  8,334,915
9 million is the 71st:  9,100,671
10 million is the 72nd: 10,347,035
```