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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.
1
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
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
KEYWORD
nonn
AUTHOR
Robin Garcia, Feb 09 2005
EXTENSIONS
More terms from Emeric Deutsch, May 31 2005
STATUS
approved