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 A090679 Integer part of the hypotenuse of a right triangle with twin prime legs. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 A212115 A091625 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

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)