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

Author

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

Keywords

• nonn
• base

Comments

(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.

Examples

			(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.
		

References

• 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

Links

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

Crossrefs

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).

Programs

Maple

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

PARI

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