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%I #12 Jun 07 2014 11:31:15
%S 5,8,17,25,42,59,84,101,144,152,195,212,254,271,280,322,339,381,398,
%T 441,492,593,610,653,738,806,848,873,907,933,1145,1162,1170,1213,1247,
%U 1442,1459,1484,1501,1544,1629,1739,1807,1824,1841,1866,2019,2053,2095,2104
%N Integer part of the hypotenuse of a right triangle with twin prime legs.
%C 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.
%H Harvey P. Dale, <a href="/A090679/b090679.txt">Table of n, a(n) for n = 1..1000</a>
%t f[n_] := IntegerPart[ Sqrt[2 n^2 + 4 n + 4]]; f[ Select[ Prime@ Range@ 250, PrimeQ[# + 2] &]] (* _Robert G. Wilson v_, Mar 10 2013 *)
%t Floor[Sqrt[Total[#]]]&/@(Select[Partition[Prime[Range[300]],2,1], Last[#]- First[#] == 2&]^2) (* _Harvey P. Dale_, Jun 07 2014 *)
%o (PARI) \Twin right triangles twinright(n) = { forprime(x=3,n, y=x+2; if(isprime(y), print1(floor(sqrt(x^2+y^2))",") ) ) }
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Dec 18 2003
%E Corrected by _T. D. Noe_, Oct 25 2006
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