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%I #20 Jun 13 2022 15:36:35
%S 5,7,53,59,151,313,1069,1789,1823,2237,2777,3329,3881,3931,4583,5227,
%T 6037,7621,7691,9467,12611,13759,14957,17609,20249,28123,35081,36979,
%U 49417,56311,56501,63857,69011,71663,79693,85439,94433,114041,117443
%N Primes p such that the concatenation of p and 29 is a square number: "p 29" = N = m^2.
%C (1) It is conjectured that the sequence is infinite.
%C (2) 29 = prime(10) is the smallest prime with the property that its digits can be the final two digits of a square.
%C (3) The possible final digits of m are necessarily e = 23, 27, 73 or 77.
%C (4) Elementary proof of (3) with (10^2 * k + e)^2 = "n 29" for these four values of e only.
%C (5) Note 23 + 77 = 27 + 73 = 10^2.
%D Andreas Bartholome, Josef Rung, Hans Kern: Zahlentheorie für Einsteiger, Vieweg & Sohn 1995
%D Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
%H Robert Israel, <a href="/A168545/b168545.txt">Table of n, a(n) for n = 1..10000</a>
%e (1) 529 = 23^2, 5 = prime(3) = a(1);
%e (2) 729 = 27^2, 7 = prime(4) = a(2);
%e (3) 5329 = 73^2, 53 = prime(16) = a(3);
%e (4) 16129 = 127^2, but 161 = 7 * 23 is composite => 161 is not a term of the sequence;
%e (5) 31329 = 177^2, 313 = prime(65) gives a(6) = 313.
%p A:= NULL:
%p count:= 0:
%p for m from 0 while count < 100 do
%p for q in [23,27,73,77] do
%p r:= floor((100*m + q)^2/100);
%p if isprime(r) then A:= A, r; count:= count+1; fi
%p od od:
%p A; # _Robert Israel_, Nov 23 2015
%o (PARI) isok(n) = isprime(n) && issquare(100*n + 29) \\ _Michel Marcus_, Jul 22 2013; corrected Jun 13 2022
%Y Cf. A000040 (the prime numbers).
%Y Cf. A167535 (concatenation of two square numbers which give a prime).
%Y Cf. A158896 (primes whose squares are a concatenation of 2 with some prime).
%K nonn,base
%O 1,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 29 2009
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