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

1,1

Comments

Conjectured to be infinite.

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.

Example

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

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 *)

Crossrefs

Cf. A036468, A002496, A005574.

Sequence in context: A136977 A003529 A018667 * A032673 A164053 A200176

Adjacent sequences: A102748 A102749 A102750 * A102752 A102753 A102754

Keyword

nonn

Author

Robin Garcia, Feb 09 2005

Extensions

More terms from Emeric Deutsch, May 31 2005

Status

approved