A102751 Numbers k such that 1 + (k-1)^2 and ((k-1)/2)^2 + ((k+1)/2)^2 = (1/2)*(k^2+1) are primes.

3, 5, 11, 15, 25, 85, 95, 121, 131, 171, 181, 205, 231, 261, 271, 315, 441, 445, 471, 545, 571, 595, 715, 751, 781, 861, 921, 951, 1011, 1055, 1081, 1095, 1125, 1151, 1185, 1315, 1411, 1421, 1495, 1615, 1661, 1701, 2035, 2051, 2055, 2065, 2175, 2261, 2315

Offset: 1

Author

Robin Garcia, Feb 09 2005

Keywords

• nonn

Comments

Conjectured to be infinite.

Examples

			11 is a term because 10^2+1=101 and 5^2+6^2=(1/2)*(11^2+1)=61 are primes.
		

References

• G. H. Hardy and W. M. Wright, Unsolved Problems Concerning Primes, Section 2.8 and Appendix 3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, p. 19.
• P. Ribenboim, The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 206-208, 1996.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Landau's Problems.

Crossrefs

Cf. A036468, A002496, A005574.

Programs

Maple

a:=proc(n) if isprime(1+(n-1)^2)=true and type((n^2+1)/2,integer)=true and isprime((n^2+1)/2)=true then n else fi end: seq(a(n),n=1..3000); # Emeric Deutsch, May 31 2005

Mathematica

Select[Range[2,2500], PrimeQ[1+(#-1)^2]&&PrimeQ[(1/2)*(#^2+1)]&] (* James C. McMahon, Jan 10 2024 *)

Extensions

More terms from Emeric Deutsch, May 31 2005