A158896 -id:A158896 - cp's OEIS frontend

A168545 Primes p such that the concatenation of p and 29 is a square number: "p 29" = N = m^2.

5, 7, 53, 59, 151, 313, 1069, 1789, 1823, 2237, 2777, 3329, 3881, 3931, 4583, 5227, 6037, 7621, 7691, 9467, 12611, 13759, 14957, 17609, 20249, 28123, 35081, 36979, 49417, 56311, 56501, 63857, 69011, 71663, 79693, 85439, 94433, 114041, 117443

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 29 2009

(1) It is conjectured that the sequence is infinite.
(2) 29 = prime(10) is the smallest prime with the property that its digits can be the final two digits of a square.
(3) The possible final digits of m are necessarily e = 23, 27, 73 or 77.
(4) Elementary proof of (3) with (10^2 * k + e)^2 = "n 29" for these four values of e only.
(5) Note 23 + 77 = 27 + 73 = 10^2.

			(1) 529 = 23^2, 5 = prime(3) = a(1);
(2) 729 = 27^2, 7 = prime(4) = a(2);
(3) 5329 = 73^2, 53 = prime(16) = a(3);
(4) 16129 = 127^2, but 161 = 7 * 23 is composite => 161 is not a term of the sequence;
(5) 31329 = 177^2, 313 = prime(65) gives a(6) = 313.

Andreas Bartholome, Josef Rung, Hans Kern: Zahlentheorie für Einsteiger, Vieweg & Sohn 1995
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040 (the prime numbers).
Cf. A167535 (concatenation of two square numbers which give a prime).
Cf. A158896 (primes whose squares are a concatenation of 2 with some prime).

A:= NULL:
count:= 0:
for m from 0 while count < 100 do
  for q in [23,27,73,77] do
    r:= floor((100*m + q)^2/100);
    if isprime(r) then A:= A, r; count:= count+1; fi
od od:
A; # Robert Israel, Nov 23 2015

isok(n) = isprime(n) && issquare(100*n + 29) \\ Michel Marcus, Jul 22 2013; corrected Jun 13 2022

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

Original entry on oeis.org

5, 53, 3329, 4583, 7691, 12611, 14957, 17609, 20249, 35081, 56501, 71663, 134909, 191231, 237851, 305477, 339539, 351293, 394007, 418997, 432569, 466079, 574109, 611993, 619841, 628373, 659831, 701741, 709469, 744251, 752903, 1386977, 1398779

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 30 2009

Subsequence of A168545: p such that p//29 = m^2.
(1) Conjecture: the sequence is infinite.
(2) 29 = prime(10) is the smallest prime which can appear as the least significant digits of perfect squares.
(3) The set of possible least significant digit pairs of m is {23, 27, 73 or 77}.
(4) Only four 2-digit primes are least significant digits of perfect squares: 29, 41, 61 and 89.
(5) There are no squares of the form p//41 = m^2 because only even numbers (no primes) concatenated with 41 are squares.

			a(1) = 5 = prime(3) because 529=23^2 and 23=prime(9).
7 = prime(4) is not in the sequence because 729=27^2 and 27=3^3 is not a prime.
a(2) = 53 = prime(16) because 5329=73^2 and 73=prime(21).
a(3) = 3329 = prime(469) because 332929=577^2 and 577=prime(106).

Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer, 4. Auflage 1998.
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005.
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.

David Corneth, Table of n, a(n) for n = 1..10000 (first 150 terms from Harvey P. Dale)

Cf. A000040, A167535, A158896, A168545.

Select[Prime[Range[120000]],PrimeQ[Sqrt[100#+29]]&] (* Harvey P. Dale, Jan 15 2019 *)

Keyword:base added by R. J. Mathar, Dec 05 2009

cp's OEIS Frontend

A168545 Primes p such that the concatenation of p and 29 is a square number: "p 29" = N = m^2.

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