Primorial primes: Difference between revisions

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#REDIRECT [[Sequence of primorial primes]]
{{draft task}}
;Definition
A [[wp:Primorial_prime|'''primorial prime''']] is a prime number that is one less or one more than a [[wp:Primorial|primorial]].

In other words a non-negative integer '''n''' corresponds to a primorial prime if either '''pn'''# - 1 or '''pn'''# + 1 is prime where '''pn'''# denotes the product of the first '''n''' primes.

;Examples
4 corresponds to the primorial prime p4# + 1 = 2 x 3 x 5 x 7 + 1 = 211.

6 corresponds to the primorial prime p6# - 1 = 210 x 11 x 13 - 1 = 30029.

;Task
Find and show here the first 12 primorial primes. As well as the prime itself show the primorial number '''pn''' to which it corresponds and whether 1 is to be added or subtracted.

As p0# (by convention) is 1 and p1# is 2, start counting from p0#.

;Stretch
If your language supports arbitrary sized integers, do the same for at least the next 12 primorial primes.

As it can take a long time to demonstrate that a large number (above say 2^64) is definitely prime, you may instead use a function which shows that a number is probably prime to a reasonable degree of certainty. Most 'big integer' libraries have such a function.

If a number has more than 40 digits, do not show the full number. Show instead the first 20 and the last 20 digits and how many digits in total the number has.

;References
* [[oeis:A228486|OEIS:A228486 - Primorial primes]]
* [[oeis:A057704|OEIS:A057704 - Primorial - 1 prime indices]]
* [[oeis:A014545|OEIS:A014545 - Primorial + 1 prime indices]]

;Related task
* [[Factorial_primes|Factorial primes]]
<br><br>
=={{header|ALGOL 68}}==
Basic task. Assumes LONG INT is at least 64 bit. Similar to the [[Factorial primes#ALGOL_68]] sample.
<lang algol68>BEGIN # find some primorial primes - primes that are p - 1 or p + 1 #
# for some primorial p #

# is prime PROC based on the one in the primality by trial division task #
PROC is prime = ( LONG INT p )BOOL:
IF p <= 1 OR NOT ODD p THEN
p = 2
ELSE
BOOL prime := TRUE;
FOR i FROM 3 BY 2 TO SHORTEN ENTIER long sqrt(p) WHILE prime := p MOD i /= 0 DO SKIP OD;
prime
FI;
# end of code based on the primality by trial divisio task #

# construct a sieve of primes up to 200 #
[ 0 : 200 ]BOOL primes;
primes[ 0 ] := primes[ 1 ] := FALSE;
primes[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO UPB primes DO primes[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO UPB primes DO primes[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( UPB primes ) DO
IF primes[ i ] THEN
FOR s FROM i * i BY i + i TO UPB primes DO primes[ s ] := FALSE OD
FI
OD;

PROC show primorial prime = ( INT p number, INT n, CHAR p op, LONG INT p )VOID:
print( ( whole( p number, -2 ), ":", whole( n, -4 )
, "# ", p op, " 1 = ", whole( p, 0 )
, newline
)
);

LONG INT pn := 1;
INT p count := 0;
INT p pos := 0;
# starting from primorial 0, which is defined to be 1 #
FOR n FROM 0 WHILE p count < 12 DO
IF LONG INT p = pn - 1;
is prime( p )
THEN
show primorial prime( p count +:= 1, n, "-", p )
FI;
IF LONG INT p = pn + 1;
is prime( p )
THEN
show primorial prime( p count +:= 1, n, "+", p )
FI;
# find the next prime #
WHILE NOT primes[ p pos +:= 1 ] DO SKIP OD;
pn *:= p pos
OD
END</lang>
{{out}}
<pre>
1: 0# + 1 = 2
2: 1# + 1 = 3
3: 2# - 1 = 5
4: 2# + 1 = 7
5: 3# - 1 = 29
6: 3# + 1 = 31
7: 4# + 1 = 211
8: 5# - 1 = 2309
9: 5# + 1 = 2311
10: 6# - 1 = 30029
11: 11# + 1 = 200560490131
12: 13# - 1 = 304250263527209
</pre>

=={{header|J}}==
Compare with the [[Factorial_primes#J|j factorial prime]] implementation.
Columns here are primorial prime number cardinal number, primorial prime, primorial index number, and offset from the corresponding primorial to the prime: <lang J> P=: 1,*/\ p:i.15 NB. primorials represented as fixed width 64 bit integers
(,.~ #\)(,. (-{&P)/"1) (,. P I. <:) /:~(#~ 1 p: ]) ,P+/1 _1
1 2 0 1
2 3 1 1
3 5 2 _1
4 7 2 1
5 29 3 _1
6 31 3 1
7 211 4 1
8 2309 5 _1
9 2311 5 1
10 30029 6 _1
11 200560490131 11 1
12 304250263527209 13 _1</lang>

=={{header|Wren}}==
===Basic===
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
<lang ecmascript>import "./math" for Int
import "./fmt" for Fmt

var limit = 12
var c = 0
var i = 0
var primes = Int.primeSieve(99) // more than enough
var p = 1
System.print("First %(limit) primorial primes:")
while (true) {
if (Int.isPrime(p-1)) {
var pi = "p%(i)#"
Fmt.print("$2d: $4s - 1 = $d", c = c + 1, pi, p - 1)
if (c == limit) return
}
if (Int.isPrime(p+1)) {
var pi = "p%(i)#"
Fmt.print("$2d: $4s + 1 = $d", c = c + 1, pi, p + 1)
if (c == limit) return
}
p = p * primes[i]
i = i + 1
}</lang>

{{out}}
<pre>
First 12 primorial primes:
1: p0# + 1 = 2
2: p1# + 1 = 3
3: p2# - 1 = 5
4: p2# + 1 = 7
5: p3# - 1 = 29
6: p3# + 1 = 31
7: p4# + 1 = 211
8: p5# - 1 = 2309
9: p5# + 1 = 2311
10: p6# - 1 = 30029
11: p11# + 1 = 200560490131
12: p13# - 1 = 304250263527209
</pre>
===Stretch===
{{libheader|Wren-gmp}}
This takes about 53.4 seconds to reach the 30th primorial prime on my machine (Core i7) with the final one taking 35 seconds of this. Likely to be very slow after that as the next primorial prime is p1391# + 1.
<lang ecmascript>import "./gmp" for Mpz
import "./math" for Int
import "./fmt" for Fmt

var limit = 30
var c = 0
var i = 0
var primes = Int.primeSieve(9999) // more than enough
var p = Mpz.one
var two64 = Mpz.two.pow(64)
System.print("First %(limit) factorial primes:")
while (true) {
var r = (p < two64) ? 1 : 0 // test for definite primeness below 2^64
if ((p-1).probPrime(15) > r) {
var s = (p-1).toString
var sc = s.count
var digs = sc > 40 ? "(%(sc) digits)" : ""
var pn = "p%(i)#"
Fmt.print("$2d: $5s - 1 = $20a $s", c = c + 1, pn, s, digs)
if (c == limit) return
}
if ((p+1).probPrime(15) > r) {
var s = (p+1).toString
var sc = s.count
var digs = sc > 40 ? "(%(sc) digits)" : ""
var pn = "p%(i)#"
Fmt.print("$2d: $5s + 1 = $20a $s", c = c + 1, pn, s, digs)
if (c == limit) return
}
p.mul(primes[i])
i = i + 1
}</lang>

{{out}}
<pre>
First 30 factorial primes:
1: p0# + 1 = 2
2: p1# + 1 = 3
3: p2# - 1 = 5
4: p2# + 1 = 7
5: p3# - 1 = 29
6: p3# + 1 = 31
7: p4# + 1 = 211
8: p5# - 1 = 2309
9: p5# + 1 = 2311
10: p6# - 1 = 30029
11: p11# + 1 = 200560490131
12: p13# - 1 = 304250263527209
13: p24# - 1 = 23768741896345550770650537601358309
14: p66# - 1 = 19361386640700823163...29148240284399976569 (131 digits)
15: p68# - 1 = 21597045956102547214...98759003964186453789 (136 digits)
16: p75# + 1 = 17196201054584064334...62756822275663694111 (154 digits)
17: p167# - 1 = 19649288510530675457...35580823050358968029 (413 digits)
18: p171# + 1 = 20404068993016374194...29492908966644747931 (425 digits)
19: p172# + 1 = 20832554441869718052...12260054944287636531 (428 digits)
20: p287# - 1 = 71488723083486699645...63871022000761714929 (790 digits)
21: p310# - 1 = 40476351620665036743...10663664196050230069 (866 digits)
22: p352# - 1 = 13372477493552802137...21698973741675973189 (1007 digits)
23: p384# + 1 = 78244737296323701708...84011652643245393971 (1115 digits)
24: p457# + 1 = 68948124012218025568...25023568563926988371 (1368 digits)
25: p564# - 1 = 12039145942930719470...56788854266062940789 (1750 digits)
26: p590# - 1 = 19983712295113492764...61704122697617268869 (1844 digits)
27: p616# + 1 = 13195724337318102247...85805719764535513291 (1939 digits)
28: p620# - 1 = 57304682725550803084...81581766766846907409 (1953 digits)
29: p643# + 1 = 15034815029008301639...38987057002293989891 (2038 digits)
30: p849# - 1 = 11632076146197231553...78739544174329780009 (2811 digits)
</pre>

Latest revision as of 10:56, 12 September 2022

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