A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

2, 10, 38, 52, 350, 542, 1102, 1460, 1522, 1732, 2510, 2642, 2768, 3692, 4592, 4658, 4690, 7238, 8180, 8320, 8960, 11392, 13468, 14920, 15908, 16600, 16832, 17878, 18820, 19100, 21532, 22060, 23240, 23842, 23968, 24622, 26428, 26638, 27170

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 04 2010

Keywords

• nonn

Comments

p = n^2 + 3, q = n^2 + 3^2 = p+6, r = n^2 + 3^3 = p+18 to be primes.

Trivially n is not a multiple of 3 and necessarily LSD of such n is e = 0, 2 or 8 as k^2+3^2 is a multiple of 5 for k = 4 or 6.

Note n^2 + m^k prime (k = 1, 2, 3) in case of m = 2 is (n^2+2,n^2+2^2,n^2+2^3) = (p,p+2,p+6): i.e., a "near square" prime triple of the first kind.

Case k=2: q is also a Pythagorean prime (A002144)

n = 350: first case where p = 122503 = prime(i), q and r are consecutive primes (i = 122503), sod(p) = sod(i) = 13, a so-called Honaker prime.

p = prime(i), q, r consecutive primes, (n,i): (350,11524) (542,25517) (1460,157987) (3692,887608) (4592,1335102) (4690,1389018).

Examples

			2^2 + 3 = 7 = prime(4), 2^2 + 3^2 = 13 = prime(6), 2^2 + 3^3 = 31 = prime(11), 2 is first term.
10^2 + 3 = 103 = prime(27), 10^2 + 3^2 = 109 = prime(29), 10^2 + 3^3 = 127 = prime(31), 10 is 2nd term.
Curiously k=0: 10^2 + 3^0 = 101 = prime(26), k=4: 10^2 + 3^4 = 181 = prime(42), necessarily LSD for such n is e = 0, k= 5: 10^2 + 3^5 = 7^3, k=6: 10^2 + 3^6 = 829 = prime(145), 10^2 + 3^7 = 2287 = prime(340), 10^2 + 3^8 = 6661 = prime(859)
n = 8180, primes for exponents k = 0, 1, 2, 3 and 4: p=66912403=prime(3946899), q=66912409=prime(3946900), r=66912427=prime(3946902), n^2+3^0=66912401=prime(3946898) and n^2+3^4=66912481=prime(3946905).
n = 8960, primes for exponents k = 1, 2, 3, 4, 5 and 6: p=80281603=prime(4684862), q=80281609=prime(4684863), r=80281627=prime(4684865), n^2+3^4=80281681=prime(4684868), n^2+3^5=80281843=prime(4684877), n^2+3^5=80282329=prime(4684904).
		

References

• F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg-Berlin 1999.
• M. du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006.

Links

• C. Hooley On nonary cubic forms, Journal für die reine und angewandte Mathematik 386, pp. 32-98, 1988.

Crossrefs

Cf. A002144, A049422, A055394, A086380, A114272.

Programs

PARI

is(n)=isprime(n^2+3) && isprime(n^2+9) && isprime(n^2+27) \\ Charles R Greathouse IV, Jun 13 2017

Extensions

More terms from R. J. Mathar, Nov 01 2010