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Definition

A *Wagstaff prime* is a prime number of the form *(2^p + 1)/3* where the exponent *p* is an odd prime.

Example

(2^5 + 1)/3 = 11 is a Wagstaff prime because both 5 and 11 are primes.

Task

Find and show here the first *10* Wagstaff primes and their corresponding exponents *p*.

Stretch (requires arbitrary precision integers)

Find and show here the exponents *p* corresponding to the next *14* Wagstaff primes (not the primes themselves) and any more that you have the patience for.

When testing for primality, you may use a method which determines that a large number is probably prime with reasonable certainty.

Note

It can be shown (see talk page) that *(2^p + 1)/3* is always integral if *p* is odd. So there's no need to check for that prior to checking for primality.

References

* Wikipedia - Wagstaff prime
* OEIS:A000979 - Wagstaff primes
assistant
```
# First 20

my @wagstaff = (^∞).grep: { .is-prime && ((1 + 1 +< $_)/3).is-prime };

say ++$ ~ ": $_ - {(1 + 1 +< $_)/3}" for @wagstaff[^20];

say .fmt("\nTotal elapsed seconds: (%.2f)\n") given (my $elapsed = now) - INIT now;

# However many I have patience for

my atomicint $count = 20;

hyper for @wagstaff[20] .. * {
    next unless .is-prime;
    say ++⚛$count ~ ": $_ ({sprintf "%.2f", now - $elapsed})" and $elapsed = now if is-prime (1 + 1 +< $_)/3;
}

```

Output:

*Turns out, "however many I have patience for" is 9 (29).*

```
1: 3 - 3
2: 5 - 11
3: 7 - 43
4: 11 - 683
5: 13 - 2731
6: 17 - 43691
7: 19 - 174763
8: 23 - 2796203
9: 31 - 715827883
10: 43 - 2932031007403
11: 61 - 768614336404564651
12: 79 - 201487636602438195784363
13: 101 - 845100400152152934331135470251
14: 127 - 56713727820156410577229101238628035243
15: 167 - 62357403192785191176690552862561408838653121833643
16: 191 - 1046183622564446793972631570534611069350392574077339085483
17: 199 - 267823007376498379256993682056860433753700498963798805883563
18: 313 - 5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
19: 347 - 95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443
20: 701 - 3506757267698915671493993255253419110366893201882115906332187268712488102864720548075777690851725634151321979237452145142012954833455869242085209847101253899970149114085629732127775838589548478180862167823854251

Total elapsed seconds: (0.09)

21: 1709 (0.87)
22: 2617 (0.92)
23: 3539 (2.03)
24: 5807 (13.18)
25: 10501 (114.14)
26: 10691 (114.14)
27: 11279 (48.13)
28: 12391 (92.99)
29: 14479 (179.87)
^C
```