A090679 Integer part of the hypotenuse of a right triangle with twin prime legs.

5, 8, 17, 25, 42, 59, 84, 101, 144, 152, 195, 212, 254, 271, 280, 322, 339, 381, 398, 441, 492, 593, 610, 653, 738, 806, 848, 873, 907, 933, 1145, 1162, 1170, 1213, 1247, 1442, 1459, 1484, 1501, 1544, 1629, 1739, 1807, 1824, 1841, 1866, 2019, 2053, 2095, 2104

Offset

1,1

Comments

The real value of these numbers is irrational. If x,x+2 are a twin prime pair then x is odd. Assume x^2 + (x+2)^2 = a^2/b^2 for integers a,b such that (a,b)=1. Since x is odd = 2m+1 we have 4m^2 + 4m + 1 + 4m^2 + 12m + 9 = 8m^2 + 16m + 10 = a^2/b^2. Multipling by b^2 we get 8m^2b^2 + 16mb^2 + 10b^2 = a^2 => a is even = 2k. So 8m^2b^2 + 16mb^2 + 10b^2 = 4k^2 or 4m^b^2 + 8mb^2 + 5b^2 = 2k. This implies b is even contrary to a being even (a,b)=1. Therefore sqrt(x^2 + (x+2)^2) is irrational and all twin prime pair legs of a right triangle form an irrational hypotenuse.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Mathematica

f[n_] := IntegerPart[ Sqrt[2 n^2 + 4 n + 4]]; f[ Select[ Prime@ Range@ 250, PrimeQ[# + 2] &]] (* Robert G. Wilson v, Mar 10 2013 *)
Floor[Sqrt[Total[#]]]&/@(Select[Partition[Prime[Range[300]], 2, 1], Last[#]- First[#] == 2&]^2) (* Harvey P. Dale, Jun 07 2014 *)

Prog

(PARI) \Twin right triangles twinright(n) = { forprime(x=3, n, y=x+2; if(isprime(y), print1(floor(sqrt(x^2+y^2))", ") ) ) }

Crossrefs

Sequence in context: A314568 A075338 A088646 * A031191 A365591 A212115

Adjacent sequences: A090676 A090677 A090678 * A090680 A090681 A090682

Keyword

easy,nonn

Author

Cino Hilliard, Dec 18 2003

Extensions

Corrected by T. D. Noe, Oct 25 2006

Status

approved