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Search: id:A090679
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| A090679 |
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Integer part of the hypotenuse of a right triangle with twin prime legs. |
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+0
1
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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.
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PROGRAM
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(PARI) \Twin right triangles twinright(n) = { forprime(x=3, n, y=x+2; if(isprime(y), print1(floor(sqrt(x^2+y^2))", ") ) ) }
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CROSSREFS
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Sequence in context: A096545 A075338 A088646 this_sequence A031191 A091625 A027601
Adjacent sequences: A090676 A090677 A090678 this_sequence A090680 A090681 A090682
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Dec 18 2003
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Oct 25 2006
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