1
$\begingroup$

Except for prime $\ p=2,\ $ primes are divided into two disjoint and about equally frequent (Dirichlet) classes of $\ p\equiv1\mod4\ $ and $\ p\equiv-1\mod4.\ $ The natural conjecture about the existence of arbitrarily long stretches, one conjecture per class, seems a bit easier than specific instances of the Schinzel Conjecture but it still feels impossibly difficult -- by a stretch I mean an interval of consecutive primes such that every two of them satisfies $\ p\equiv q\mod 4.$

Question 1:   What is (approximately) the smallest prime that initiates a stretch of length n? (primes $\ P_1(n)\ $ and $\ Q_3(n)\ $ for each of the two classes respectively).

Let $\ \ P_1(n)\equiv1\!\!\mod4\ $ and $\ \ Q_3(n)\equiv3\!\!\mod4\ $ be these two parameters. Thus

$$ P_1(1)=5\qquad\text{and}\qquad Q_3(1)=3; $$

$$ P_1(2)=13\qquad\text{and}\qquad Q_3(2)=7; $$

and the numerical challenge gets harder and harder. It's especially interesting how the two classes differ in the given context.

===================

A pulsar is a finite interval of consecutive odd primes within which $\ p_k\not\equiv p_{k+1}\mod 4$. Perhaps it's utmost hard to prove the existence of arbitrarily long pulsars. Thus let me ask

Question 2:   What is (approximately) the smallest prime $\ A(n)\ $ that initiates a pulsar of length n?

For instance,

$$ A(1)=A(2)=A(3)=3 $$

====================

Finally, one would like to compare stretches and pulsars.

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ oeis.org/A055623 tabulates "First occurrence of run of primes congruent to $1$ mod $4$ of exactly length $n$." oeis.org/A055624 does the same for $3$ mod $4$. $\endgroup$
    – Gerry Myerson
    Commented Jul 10, 2021 at 4:09
  • 1
    $\begingroup$ oeis.org/A247384 tabulates "Find the first (maximal) string of consecutive primes of length exactly $n$ which alternate between $4k+1$ and $4k+3$ or $4k+3$ and $4k+1$ as in A002144($4n+1$) and A002145($4n+3$)." I don't understand why it starts with $97$ and not with $7$. oeis.org/A289118 tabulates "Least prime beginning a string, of length at least $n$, of consecutive primes which alternate between types $4k+1$ and $4k+3$ or $4k+3$ and $4k+1$." $\endgroup$
    – Gerry Myerson
    Commented Jul 10, 2021 at 4:19
  • 1
    $\begingroup$ @GerryMyerson, thank you for both links. $97$ must be the first isolated prime, meaning that it is congruent to the previous, $\ 89,\ $ and to the next prime, $\ 101\mod4$ $\endgroup$
    – Wlod AA
    Commented Jul 10, 2021 at 4:30
  • $\begingroup$ @GerryMyerson, I closed my eyes and checked primes from 2 to 97, and I agree with oeis about 97. $\endgroup$
    – Wlod AA
    Commented Jul 10, 2021 at 4:37
  • $\begingroup$ OEIS records the exact stretches (nice!). The sequences are not increasing -- interesting. $\endgroup$
    – Wlod AA
    Commented Jul 10, 2021 at 4:47

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .