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A178545 Primes p such that q = p^2 + p + 1 is an emirp.

%I A178545 #18 Aug 13 2025 22:48:07
%S A178545 3,5,41,59,839,857,1811,1931,3011,3221,3407,3671,8387,8543,8627,9719,
%T A178545 9743,9803,10781,11549,12647,13469,13487,13499,13613,13931,14087,
%U A178545 17477,17573,17837,18089,18269,19319,19403,19661,19991,27191,27947,31223,33311,34313
%N A178545 Primes p such that q = p^2 + p + 1 is an emirp.
%C A178545 It is conjectured (but still an open problem) that there exist infinitely many primes of the form n^2 + n + 1 = ((2*n+1)^2 + 3)/4.
%C A178545 Landau's 4th problem from (1912, 5th Congress of Mathematicians in Cambridge) conjectures that there are infinitely many primes of the form n^2 + 1 (also Euler 1760; Mirsky 1949).
%C A178545 Hardy and Littlewood proposed a conjecture about the asymptotic number of primes of the form n^2 + 1.
%C A178545 An emirp ("prime" spelled backwards) is a prime whose reversal is a different prime, the reversal of q is denoted by R(q).
%C A178545 It is conjectured but also unproved that there are infinitely many emirps (see A048054).
%C A178545 For p > 3 necessarily p of the form 6*k + 5 as (6*k+1)^2 + (6*k+1) + 1 a multiple of 3.
%D A178545 M. Gardner: Die magischen Zahlen des Dr. Matrix, Krueger Verlag, Frankfurt am Main, 1987
%D A178545 R. Guy: Unsolved Problems in Number Theory,3rd edition, Springer, New York, 2004
%D A178545 G. H. Hardy, E. M. Wright: Einfuehrung in die Zahlentheorie, R. Oldenburg, Muenchen, 1958
%H A178545 Robert Israel, <a href="/A178545/b178545.txt">Table of n, a(n) for n = 1..10000</a>
%e A178545 3^2 + 3 + 1 = 13 = prime(6), R(13) = prime(11), 3 is first term.
%e A178545 5^2 + 5 + 1 = 31 = prime(11), R(31) = prime(6), 5 is 2nd term.
%e A178545 q = 1811^2 + 1811 + 1 = 3281533 = prime(235691), R(q) = prime(240351), first case that p = 1811 = prime(280) = emirp(87) is itself an emirp.
%p A178545 filter:= proc(p) local q,qr;
%p A178545    if not isprime(p) then return false fi;
%p A178545    q:= p^2+p+1;
%p A178545    if not isprime(q) then return false fi;
%p A178545    qr:= revdigs(q);
%p A178545    qr <> q and isprime(qr);
%p A178545 end proc:
%p A178545 select(filter, [3,seq(i,i=5..50000,6)]); # _Robert Israel_, Dec 04 2016
%t A178545 EmirpQ[n_] := If[ PrimeQ@n, Block[{id = IntegerDigits@n}, rid = Reverse@ id; rid != id && PrimeQ@ FromDigits@ rid]]; Select[ Prime@ Range@ 3700, EmirpQ[ #^2 + # + 1] &] (* _Robert G. Wilson v_, Jul 26 2010 *)
%t A178545 p2emrpQ[p_]:=With[{q=p^2+p+1},!PalindromeQ[q]&&AllTrue[{q,IntegerReverse[q]},PrimeQ]]; Select[Prime[Range[3700]],p2emrpQ] (* _Harvey P. Dale_, Mar 10 2025 *)
%Y A178545 Cf. A000040, A002383, A048054, A006567, A109308, A109309.
%K A178545 base,nonn,look
%O A178545 1,1
%A A178545 Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010
%E A178545 More terms from _Robert G. Wilson v_, Jul 26 2010