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

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

1,1

Comments

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.

References

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.

Links

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

Example

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).

Mathematica

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

Crossrefs

Cf. A000040, A167535, A158896, A168545.

Sequence in context: A099086 A304999 A106045 * A048551 A236529 A083757

Adjacent sequences: A168565 A168566 A168567 * A168569 A168570 A168571

Keyword

nonn,base

Author

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

Extensions

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

Status

approved