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A090682 Integer part of one leg of a right triangle where the other leg and hypotenuse are twin primes. 0
4, 4, 6, 8, 10, 12, 15, 16, 20, 20, 23, 24, 26, 27, 28, 30, 30, 32, 33, 35, 37, 40, 41, 42, 45, 47, 48, 49, 50, 51, 56, 57, 57, 58, 59, 63, 64, 64, 65, 66, 67, 70, 71, 71, 72, 72, 75, 76, 76, 77, 80, 80, 81, 82, 82, 84, 86, 86, 87, 88, 89 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Except for the first term, the real value of these numbers is irrational. We have for prime p>2, (p+2)^2 - p^2 = a^2 or 4p+4 = a^2. This implies a=4k. and 4p+4 = 16k^2 or p = 4k^2-1 = (2k+1)(2k-1) = composite for k > 1. Therefore for a = 4, p=3 is the only integer solution. Suppose 4p+4 = a^2/b^2 for (a,b) =1. 4pb^2 - 4b^2 = a^2 = 16k^2 or pb^2 = 4k^2 - b^2 = (2k-b)(2k+b) => k=br. Then pb^2 = (2br-b)(2br+b) or p = (2r-1)(2r+1) and p is prime only r = 1. So pb^2 = (b)(3b) and p = 3 is the smallest and only rational solution.

LINKS

Table of n, a(n) for n=3..63.

PROG

(PARI) \Twin right triangles twinright2(n) = { forprime(x=3, n, y=x+2; if(isprime(y), print1(floor(sqrt(y^2-x^2))", ") ) ) }

CROSSREFS

Sequence in context: A200267 A273308 A105790 * A011982 A282939 A175216

Adjacent sequences:  A090679 A090680 A090681 * A090683 A090684 A090685

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, Dec 18 2003

STATUS

approved

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Last modified October 14 06:58 EDT 2019. Contains 327995 sequences. (Running on oeis4.)