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.
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A pulsar is a finite interval of consecutive odd primes within which $\ p_k\not\equiv p_{k+1}\mod 4$\ 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,
$$ A(1)=A(2)=A(3)=3 $$
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Finally, one would like to compare stretches and pulsars.
