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

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

Author

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

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

• 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.

Crossrefs

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

Extensions

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