Problems & Puzzles: Puzzles

Puzzle 539. n consecutive primes same residue modulo 4 Anton Vrba sent the following puzzle: Find n consecutive primes P1, P2 ... Pn all having the same residue modulo 4 and n larger than ln(P1). We define Q = n / ln(P1)    7 consecutive primes 3 mod 4, Q=1.14049, P1=463 13 consecutive primes 3 mod 4, Q=1.04881, P1=241,603 21 consecutive primes 1 mod 4, Q=1.00813, P1=1,113,443,017 22 consecutive primes 1 mod 4, Q=1.05613, P1=1,113,443,017 23 consecutive primes 1 mod 4, Q=1.10414, P1=1,113,443,017 24 consecutive primes 1 mod 4, Q=1.15214, P1=1,113,443,017   Q1:  Can you extend the list?

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Contributions came from J. K. Andersen & Maximilliam Hasler

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Andersen wrote:

The first run of exactly n consecutive primes congruent to 1 mod 4
and congruent to 3 mod 4 is at
http://users.cybercity.dk/~dsl522332/math/congruent-primes.htm#mod4

The cases with n larger than ln(P1):
2 consecutive primes 3 mod 4, Q=1.02780, P1=7
7 consecutive primes 3 mod 4, Q=1.14049, P1=463
13 consecutive primes 3 mod 4, Q=1.04881, P1=241,603
24 consecutive primes 1 mod 4, Q=1.15214, P1=1,113,443,017
22 consecutive primes 1 mod 4, Q=1.04184, P1=1,481,666,377
22 consecutive primes 3 mod 4, Q=1.03271, P1=1,786,054,147
23 consecutive primes 1 mod 4, Q=1.06147, P1=2,572,421,893
26 consecutive primes 1 mod 4, Q=1.03330, P1=84,676,452,781
32 consecutive primes 3 mod 4, Q=1.25716, P1=113,391,385,603
29 consecutive primes 1 mod 4, Q=1.06388, P1=689,101,181,569
33 consecutive primes 1 mod 4, Q=1.14510, P1=3,278,744,415,797
30 consecutive primes 1 mod 4, Q=1.04088, P1=3,289,884,073,409
31 consecutive primes 3 mod 4, Q=1.05116, P1=6,425,403,612,031
30 consecutive primes 3 mod 4, Q=1.00629, P1=8,858,854,801,319

This is not exhaustive for the puzzle. Some of the above P1 also
qualify for smaller n, for example 113,391,385,603 for n = 26 to 31.
Some unlisted P1 would qualify but are not the first run for that n.

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Maximilliam Hasler wrote:

Concerning http://www.primepuzzles.net/puzzles/puzz_539.htm,
some more values are P1 = [84676452781, 689101181569, 3289884073409, 3278744415797] with Q = [1.033299942945145055521885426,  1.063882319839893777615605035, 1.040876126884962411487954260, 1.145098495598602519112185053] and especially (this one ==3 mod 4 !)
Q( p=113 391 385 603 ) = 32/log(p) = 1.25716430137


Relevant sequences from OEIS are
A055623                  First occurrence of run of primes congruent to 1 mod 4 of
exactly length n.
A054678                  n consecutive primes differ by a multiple of 4 starting at a(n).
A057624                  Initial prime in first sequence of n primes congruent to
1 modulo 4.
and references therein (esp. JKA's page) ; see
http://www.research.att.com/~njas/sequences/?q=1113443017

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