

A102751


Numbers n such that 1+(n1)^2 and ((n1)/2)^2+((n+1)/2)^2=(1/2)*(n^2+1) are primes.


0



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
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OFFSET

1,1


COMMENTS

Conjectured to be infinite.
We still do not know if there are an infinite number of primes of the form n^2 + 1, which is Landau's 4th Problem, declared "unattackable" in 1912. See Hardy and Wright; Ribenboim.  Jonathan Vos Post, Feb 28 2005


REFERENCES

Hardy, G. H. and Wright, W. M., 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.
Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: SpringerVerlag, pp. 206208, 1996.


LINKS

Table of n, a(n) for n=1..49.
Eric Weisstein's World of Mathematics, Landau's Problems.


EXAMPLE

a(3)=11 because 10^2+1=101 and 5^2+6^2=(1/2)*(11^2+1)=61 are prime


MAPLE

a:=proc(n) if isprime(1+(n1)^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


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



