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Old 2018-07-06, 19:38   #1
carpetpool
 
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"Sam"
Nov 2016

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Post OEIS A213052 for primes p = 3 mod 4

The OEIS sequence A213052 is a sequence of primes p such that all primes less than the nth prime are primitive roots mod p. Except for 3, all the primes in the sequence are congruent to 5 mod 24 and therefore congruent to 1 mod 4. What would make this an interesting problem is finding the sequence of smallest primes p = 3 mod 4 for which all primes less than the nth prime are primitive roots mod p. Also, we could consider a sequence of smallest primes p = 3 mod 4 such that for all primes q less than the nth prime, -q is a primitive root mod p.


For the first case they are:


3, 19, 907, 907, 2083, 101467, 101467, 350443,



and the second case:


7, 23, 239, 479, 479, 1559, 9239, 10559,



Is anyone able to find more terms for any of the two sequences?
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Old 2018-07-11, 22:27   #2
danaj
 
"Dana Jacobsen"
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Using a simple modification of the Perl script on the OEIS page (just add the $p % 4 == 3 clause), I get:

3, 19, 907, 1747, 2083, 101467, 350443, 916507, 1014787, 6603283, 27068563, 45287587, 226432243, 243060283, 3946895803, 5571195667, 9259384843

The Perl code is about 7x faster than Pari on my computer, and got the above in 10 minutes.


For your second sequence:

7, 23, 239, 479, 1319, 1559, 9239, 10559, 35279, 250799, 422231, 701399, 4080359, 13147679, 13518119, 48796439, 94123559, 102628679, 120293879, 149013479, 688333799, 1595386679, 2929911599

Note that A213052 indicates it is an increasing sequence of primes, hence we don't have repeats. To get the repeats change the 'if' to a 'while', which gives 479 instead of 1319; 4080359, 120293879, and 2929911599 twice each.
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Old 2018-07-14, 17:00   #3
carpetpool
 
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"Sam"
Nov 2016

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Quote:
Originally Posted by danaj View Post
Using a simple modification of the Perl script on the OEIS page (just add the $p % 4 == 3 clause), I get:

3, 19, 907, 1747, 2083, 101467, 350443, 916507, 1014787, 6603283, 27068563, 45287587, 226432243, 243060283, 3946895803, 5571195667, 9259384843

The Perl code is about 7x faster than Pari on my computer, and got the above in 10 minutes.


For your second sequence:

7, 23, 239, 479, 1319, 1559, 9239, 10559, 35279, 250799, 422231, 701399, 4080359, 13147679, 13518119, 48796439, 94123559, 102628679, 120293879, 149013479, 688333799, 1595386679, 2929911599

Note that A213052 indicates it is an increasing sequence of primes, hence we don't have repeats. To get the repeats change the 'if' to a 'while', which gives 479 instead of 1319; 4080359, 120293879, and 2929911599 twice each.
How did I miss 1747 in the first (and more important) case?
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