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A168568 Primes p such that the concatenation p//29 is a squared prime.

%I A168568 #22 Jul 30 2025 00:58:43
%S A168568 5,53,3329,4583,7691,12611,14957,17609,20249,35081,56501,71663,134909,
%T A168568 191231,237851,305477,339539,351293,394007,418997,432569,466079,
%U A168568 574109,611993,619841,628373,659831,701741,709469,744251,752903,1386977,1398779
%N A168568 Primes p such that the concatenation p//29 is a squared prime.
%C A168568 Subsequence of A168545: p such that p//29 = m^2.
%C A168568 (1) Conjecture: the sequence is infinite.
%C A168568 (2) 29 = prime(10) is the smallest prime which can appear as the least significant digits of perfect squares.
%C A168568 (3) The set of possible least significant digit pairs of m is {23, 27, 73 or 77}.
%C A168568 (4) Only four 2-digit primes are least significant digits of perfect squares: 29, 41, 61 and 89.
%C A168568 (5) There are no squares of the form p//41 = m^2 because only even numbers (no primes) concatenated with 41 are squares.
%D A168568 Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer, 4. Auflage 1998.
%D A168568 Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005.
%D A168568 Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
%H A168568 David Corneth, <a href="/A168568/b168568.txt">Table of n, a(n) for n = 1..10000</a> (first 150 terms from Harvey P. Dale)
%e A168568 a(1) = 5 = prime(3) because 529=23^2 and 23=prime(9).
%e A168568 7 = prime(4) is not in the sequence because 729=27^2 and 27=3^3 is not a prime.
%e A168568 a(2) = 53 = prime(16) because 5329=73^2 and 73=prime(21).
%e A168568 a(3) = 3329 = prime(469) because 332929=577^2 and 577=prime(106).
%t A168568 Select[Prime[Range[120000]],PrimeQ[Sqrt[100#+29]]&] (* _Harvey P. Dale_, Jan 15 2019 *)
%Y A168568 Cf. A000040, A167535, A158896, A168545.
%K A168568 nonn,base
%O A168568 1,1
%A A168568 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 30 2009
%E A168568 Keyword:base added by _R. J. Mathar_, Dec 05 2009