A176371 -id:A176371 - cp's OEIS frontend

A176969 Numbers n such that n^2 + 13^2 is prime.

2, 8, 10, 12, 20, 22, 28, 30, 32, 38, 42, 48, 58, 60, 62, 68, 80, 90, 100, 108, 110, 112, 122, 128, 138, 142, 148, 150, 168, 172, 180, 190, 198, 200, 202, 210, 228, 230, 232, 238, 242, 248, 258, 262, 268, 280, 282, 302, 310, 318, 340, 342, 360, 362, 368, 378

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 29 2010

nonn

The n^2 + d conjecture is a famous and still unsolved problem.
It states that there exist an infinite number of primes whose values are of the form n^2 + d for some integer n.
This is case d = 13^2.

			2^2 + 13^2 = 173 = prime(40), 2 is first term.
12^2 + 13^2 = 313 = prime(65) = palprime(11), 12 is 4th term.
310^2 + 13^2 = 96269 = prime(9274) = palprime(106), 310 the 49th term.

J. Matousek, J. Nesetril: Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007
M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006

Cf. A000040, A000290, A056899, A113536, A176371.

isok(n) = isprime(n^2 + 13^2) \\ Michel Marcus, Jun 28 2013

A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

Original entry on oeis.org

2, 10, 12, 28, 42, 60, 68, 108, 110, 122, 142, 180, 198, 200, 268, 282, 340, 378, 380, 418, 430, 488, 500, 502, 530, 612, 742, 788, 802, 838, 840, 912, 942, 948, 952, 1010, 1080, 1102, 1148, 1232, 1270, 1428, 1452, 1472, 1502, 1522, 1538, 1720, 1778, 1830

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 30 2010

nonn

Numbers are subsequence of A113536 and A176969

See comments and references of A176969

			2^2 + 13 = 17 = prime(7), 2^2 + 13^2 = 173 = prime(40), 2 is first term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000290, A056899, A113536, A176371, A176969.

Select[Range[2000],AllTrue[#^2+{13,13^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2015 *)

isok(n) = isprime(n^2 + 13) && isprime(n^2 + 13^2); \\ Michel Marcus, Aug 27 2013

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

Original entry on oeis.org

10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010

nonn

Subsequence of A176969.

The least-significant digit of all terms is one of 0, 2 or 8, because for odd digits m^2 + 13^k would be even (not prime), and for digits 4 and 6 the number m^2 + 13^2 is a multiple of 5.

			m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.

B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.

Cf. A000040, A000290, A055394, A056899, A086380, A113536, A176371, A176969.

keyword:base removed by R. J. Mathar, Jul 13 2010

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A176969 Numbers n such that n^2 + 13^2 is prime.

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A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

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A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

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